What underpins this difference? Does the US legal framework provide
more opportunity for such challenges? Is the US electoral process more
amenable to fraud? Or is there simply more appetite for litigation in
I'll add a few fairly important basic points which the other answers don't explicitly mention. There are many reasons litigation is more attractive in the U.S.
Also, the amount of election litigation in the U.K. as another answer notes, is not zero. (A more comprehensive list of U.K. election lawsuits from the comments to the question can be found here).
Lists of U.S. election contests in U.S. Senate races, Presidential elections, and other U.S. elections (with some overlap and a non-comprehensive list). Election outcome disputes and litigation were much more common in federal elections prior to the U.S. Civil War than they are today.
Short Ballots v. Long Ballots
U.K. elections are profoundly more simple than U.S. elections in terms of logistics.
In a U.K. election voters are typically casting a single vote for a single office (or a single rare ballot issue) at any one time on their ballot. This happens once, without a preceding primary election. The person who gets the most votes wins. When it happens, everyone in the country, and the media in the entire country, is focused on that race, and ill informed voters can rely in partisan cues to decide how to vote.
In contrast, a typical U.S. ballot is very long. My ballot this year is typical. I have 47 different contests to vote upon.
The number of issues that can come up in a 47 issue ballot are myriad compared to the number of issues that can come up in a single candidate race ballot. This was preceded a few months earlier by one or more primary elections (this year, Colorado actually had a Presidential primary, a partisan caucus and a non-Presidential primary).
The length of a typical U.S. ballot also means that counting procedures pretty much have to be automated the first time around and are far more technical. So there is far more to dispute regarding how ballots are actually counted.
Further, recounts are needed if any race on the long ballot is close, not just a single race. So, there may need to be a recount even if the top of the ticket race isn't close. Most years some race, somewhere in a state will need to be recounted.
Low Information Voters and Election Fatigue
One of the other features of the long ballots and complicated political system of the U.S. is that voters typically have lots of things to vote upon that haven't received much media attention and lots of voters who don't understand everything they are voting upon.
There are lots of non-partisan as well as partisan candidate races, lots of obscure offices decided in down ticket races, and lots of complex ballot issues and judicial elections.
It also is worth noting that the U.S. has more internal mobility than the U.K. so there are more people in any jurisdiction at any given time who are unfamiliar with the local process (something exacerbated by the fact that the process and issues and candidates to vote up differ a lot from place to place in the U.S.).
So, in the U.S. there are lots of low information and confused voters in some races which makes those races especially vulnerable to even slight manipulations, so the returns to litigating slight nuances of voting procedures are much greater in the U.S. because voters are easier to sway in many races.
A related point is that because there are so many elections for so many things in the U.S., many of which are for low stakes, turnout tends to be lower and for this and other reasons, small changes can have big impacts on turnout. If you have one election that takes very little work for the voters, every five years, with unmistakably high stakes, turnout tends to be higher. Lots of election litigation in the U.S. is related to turnout suppression.
More Elections To Litigate
In the U.S. most jurisdictions have at least four general elections and two primary elections every four years and often more, ignoring local municipal elections. The U.K. has one general election irregularly but roughly every five years for elections above the municipal level.
There are also more kinds of local government elections in the U.S. than in the U.K., which are held at a variety of times in a variety of ways. Denver, where I live is fairly simple, yet has three different procedures for voting for different local officials (at large non-partisan, single district partisan and single district non-partisan) in the city alone, and three governmental entities serving only the city proper with elected officials in them (school board, city and DA). Most places have more complex local government elections.
More elections create more opportunities for disputes and litigation.
Fragmented v. Consolidated Election Administration and Partisanship
U.S. federal elections for President are administered by 51 different election administration systems (one for each state plus one for the Presidential election in the District of Columbia).
In the vast majority of those jurisdictions, election administration is lead by a partisan elected official, usually called a Secretary of State, which in turn leads dozens of local partisan elected officials, usually called County Clerks, some of whom differ in party from the Secretary of State. Sometimes bipartisan election boards are used instead.
The fact that elections are administered by partisan elected officials in the U.S. greatly increases the likelihood that an elected official will push the boundaries of what is legally allowed to the legal limit and creates suspicion which lawyers seek to address when suspicions may cross a legal line.
Also, this means that not just each of 51 larger jurisdictions of election administration, but also the roughly 3000 county level election administrators, are largely autonomous of each other and may conduct the election in a non-uniform manner that influences the result.
Even once you get the results, not every jurisdiction counts ballots the same way in federal elections. Some have British style single member district plurality votes. Maine has ranked choice voting for House and Senate seats, and splits its electoral college votes up by Congressional district (as does Nebraska). Some states require majorities of votes cast to win and hold a runoff if there isn't a winner in the first round. The voting or counting machines used likewise differ, often even within a single state.
Sometimes, a by-election to fill a vacancy is held at the same time as an ordinary election for the U.S. Senate seat so you have two parallel races for both of a state's Senate seats at the same time, something that basically never happens in the U.K. where by-elections to fill vacancy are usually held promptly and separately, and all MPs are elected at the same time (like the U.S. House).
The fact that eligibility to vote is handled by a single consolidated national bureaucracy in the U.K., rather than fragmented as in the U.S., also means that there is much less room for dispute over who is entitled to vote where, and less room for inconsistent procedures regarding voting qualification to arise.
Lots of Kinds of Voting
Unlike the U.K., in the U.S., elections are no longer simple one day, in-person affairs. Most states have some combination of absentee voting, special rules for military voting, mail in ballots (other than need based absentee voting), early in-person voting and election day voting. Therefore, the typical U.S. election is in the process of being conducted for weeks, not hours.
A History of Race Based Election Trickery
The U.S. enacted the 1964 Voting Rights Act because, prior to that date, abuses by state officials in election administration, and other forms of unfair election trickery were common place, usually motivated by a desire to discriminate because race and partisan preference are closely tied in the U.S.
This has been tamed in the meantime, but has never completely ended as a partisan tactic. Lawyers are deployed to keep that kind of conduct in check.
While the U.K. has lots of ethnic diversity now, historically it had a long time to develop norms for conducting elections that preceded the appearance of this diversity and it never had widespread "party machine" politics driven by large, ill informed immigrant populations of voters as the U.S. did historically.
Judicial System Differences
Due to federalism, the court system in the U.S. is vastly more complex, as applied to election disputes, than the court system in the U.K.
In the U.S. there are parallel state and federal courts, typically with one state court with jurisdiction over election disputes in each county in which they might arise, and one or more federal trial courts in each state. The state and federal courts are supervised by separate appellate courts, and the law governing which court handles which kind of case is intricate, technical and complicated. Half of a typical election law case is typically just figuring out who can bring what kind of issue in which court of many options.
Furthermore, the U.S. courts are much more partisan in appointment and in how they handle public law issues, than the U.K. courts where judges are basically just one more class of civil servants.
U.S. courts (not just the U.S. Supreme Court, but every court below it) also have the power to declare laws invalid due to a constitutional violation or a state statute that violates federal law, which is something that U.K. courts generally can't do, at least in election cases, which gives courts in the U.S. more power and flexibility and makes them a more attractive place to litigate than U.K. courts for someone concerned about the status quo outcome.
Clarity of Laws Due To Separation Of Powers And Bicameralism
Because the U.K. almost never has legislation enacted by a divided legislature or a head of government who belongs to a different party than the legislature which supports the head of government that called the election, there is no incentive for U.K. legislators to write unclear laws and the people charged with administering the laws are more involved in drafting them than in the U.S.
In the U.S., statutes are often intentionally vague or ambiguous, since clarity might not secure bipartisan majorities needed to enact laws, and the status quo laws on the books are much harder for an incumbent election administrator to change most of the time, so there are more hard questions for judges to decide. Judges are trusted more than legislators most of the time, so this can be politically prudent. So, U.S. election laws often leave more legitimate issues for disagreement than U.K. election laws.
Also, due to federalism, U.S. election laws have more components - the U.S. Constitution, federal statutes, federal regulations, state constitutions, state statutes, state regulations and county regulations all come into play and have to be interpreted together, rather than there being one authoritative blueprint for legal guidance that applies everywhere.
Further, precedents applying on state's election laws aren't binding when applying another state's election laws, so the precedent system resolves ambiguities in the law more quickly in the U.K. than in the U.S.
Big Direct Elections
In the U.S. the largest jurisdictions where direct vote counts matter are much larger than the largest MP district (California has more than half of the population of the entire U.K. for example, and is only one of 51 federal election administration jurisdictions). So, a statewide recount in a large state like Florida is much more of an enterprise than a MP district wide recount.
There are perhaps 40,000-50,000 votes cast in each constituency, in a typical U.K. election. The typical U.S. House race has 770,000 people and hundreds of thousands of votes cast. The median state has several million residents and at least a million votes cast in a federal election, all of whose votes must be aggregated at the state level in a U.S. Senate or Presidential race (or counted by both Congressional District and at the State level simultaneously in Maine and Nebraska).
And, except in elections so close that one seat will change the majority in parliament, losing a close race for an MP won't change the balance of power. In contrast, in the U.S., recounts can often change the balance of power because there is a quite even U.S. Senate and Electoral College default partisan divide in the U.S. with state sized chunks and some of the perennial swing states are large in population.
Recounting 1% of close MP races is much easier than recounting several large U.S. states and this leaves more to fight about. Each individual MP count is also in a logistically more compact area than each individual U.S. state. In a related matter, U.S. election districts are less closely aligned with political boundaries of election administration entities (in part because equal population requirements in redistricting are quite strict in the U.S., unlike other considerations), so typically any given federal race may have dozens of different election administrators under the supervision of different elected partisan election administration officials to merge, which is more rare and less extreme in the U.K.
Both the U.S. and the U.K. have gerrymandering.
In the U.S. there is some room to litigate the fairness of gerrymandered districts in advance. In the U.K., this is less feasible on the merits in light of the structure of U.K. election law (but there is less extreme gerrymandering done).
Once gerrymandering has happened, it tends to make individual district races less competitive since most districts are drawn in a way that by chance or by design, favors one candidate or another strongly.
In the U.S., however, state boundaries which govern the shape of district boundaries for U.S. Senate and Presidential elections can't be changed for all practical purposes. So, in those races, the districts have not been intentionally drawn to make as many states as possible safe districts. Some states have just one U.S. House district which can't be gerrymandered either.
The end result is that are proportionately more people in the U.S. who are located in swing states for some race (even if not for all races), than there are people, proportionately, in the U.K. who are in swing constituencies. And, since only close races make sense to litigate, there is more election litigation in the U.S. for this reason as well.
You would think that the stakes would be higher in the U.K. when you transfer complete control of the government in each election. And that does drive up turnout.
But while the winner gets a bigger prize in the U.K., the differences between the likely coalitions in the U.K. based on an election, and the differences in the U.S., are in my humble opinion, less extreme. You can see this in the chart below which doesn't even control for the moderating effect of multi-party coalitions that is often present in the U.K. parliament:
While the Labour Party in Britain is slightly to the left of the U.S. Democratic Party, the Conservative Party in Britain is far more moderate than the U.S. Republican Party.
A change in control in the U.S. is between two parties with quite an extreme left-right divide, while my perception is that there is more of a national consensus on a lot of core political decisions (e.g. keeping national health care in place) that reduces the pain of losing when you do lose.
Put another way, I perceive the U.K. to be less deeply culturally and politically divided than the U.S.
Also, the U.K. has fewer circumstances in which in incumbent political party can entrench political outcomes that future elections can't easily undo, in the way that U.S. Supreme Court appointments, decennial redistricting and gerrymandering, or even ordinary legislation which is hard to change with divided government and a U.S. Senate filibuster, can in the U.S.
When the two sides have little in common, the desire to fight every battle you can to the greatest extreme possible is greater.
Some of these considerations apply in other countries, some don't. But, few other countries have as long a list of factors favoring litigation as the U.S. does.
Most have shorter ballots, higher information voters with less election fatigue (manipulation of low turnout is not nearly so great a factor almost anywhere else in the world beside the U.S.), fewer elections, consolidated non-partisan election administration, fewer kinds of voting, less of a history of election trickery, a more unified court system with less opportunity for judicial review, and a parliamentary system which tends to result in clearer legislative drafting.
Many have less stark cultural and political divisions (e.g. Japan), but some countries have divisions just as deep or deeper (e.g. Nigeria or the Ukraine). Some also have large direct jurisdiction direct elections, just as the U.S. does, or even larger ones with national direct elections for a President or on a ballot issue.
Gerrymandering is rare elsewhere due to proportional representation which is common in many places. Proportional representation can make vote counting more complex and can make irregularities in places where one party is strongly favored more salient, but prevents any one irregularity from having hugely outsized consequences most of the time.
Also, in comparisons other than the U.S. and the U.K., the available supply of lawyers is a much bigger factor in election litigation rates. While these countries have ample supplies of lawyers to litigate elections, in many countries lawyers who can litigate elections are far more scarce and that can reduce election litigation. The ample supply of lawyers in the U.S. certainly doesn't discourage high levels of election litigation.