Short Answer
In a nutshell, aristocrats, and their extended families who are not aristocrats, make up an outsized share of the power and wealth elites of modern Britain. But, they have largely lost the formal rights and privileges that they had as recently as the Victorian era and early 20th century.
They now share their educational, economic and power privileges with wealthy commoners who derive their wealth from professional and business activities. they now share their power with members of parliament, regional legislators, municipal politicians, senior judges, and senior civil servants, who are disproportionately derived from this elite, but also include commoners of much more humble origins.
The wealth of the aristocracy in Britain peaked in the late 1800s, took hits in World War I and World War II, hit a low point in the last 1970s and early 1980s. But since then, it has significantly rebounded to near record highs in inflation adjusted dollars, although aristocrats are still not quite as dominant in their land holdings as they were in the Victorian era and earlier times.
Simple inherited wealth has proven more enduring for Britain's peers than inherited legal and political privileges.
Formal Privileges Of Peerage
Prestige, an exclusive right similar to a trademark right to use the aristocratic title, and eligibility to serve in the House of Lords if elected by one's fellow aristocrats to the remaining posts for hereditary lords in the body, are the main remaining privileges of British aristocrats.
A few of the rights historically shared by all peers, however, are still retained by the Queen, and sometimes also by some royals, and sometimes by serving members of the House of Lords.
The privilege of peerage is the body of special privileges belonging
to members of the British peerage. It is distinct from parliamentary
privilege, which applies only to those peers serving in the House of
Lords and the members of the House of Commons, while Parliament is in
session and forty days before and after a parliamentary session.
The privileges have been lost and eroded over time. Only three
survived into the 20th century: the right to be tried by other peers
of the realm instead of juries of commoners, freedom from arrest in
civil (but not criminal) cases, and access to the Sovereign to advise
him or her on matters of state. The right to be tried by other
peers was abolished in 1948. Legal opinion considers the right of
freedom from arrest as extremely limited in application, if at
all. The remaining privilege is not exercised and was
recommended for formal abolition in 1999, but has never been formally
revoked.
Peers also have several other rights not formally part of the
privilege of peerage. For example, they are entitled to use coronets
and supporters on their achievements of arms.
. . .
Now peers are tried by juries composed of commoners, though peers were
themselves excused from jury service until the House of Lords Act 1999
restricted this privilege to members of the House of Lords. The right
to be excused was abolished on 5 April 2004 by the Criminal Justice
Act 2003.
(Peers were, and still are hypothetically, subject to impeachment.
Impeachment was a procedure distinct from the aforementioned procedure
of trial in the House of Lords, though the House of Lords is the court
in both cases. Charges were brought by the House of Commons, not a
grand jury. Additionally, while in normal cases the House of Lords
tried peers only for felonies or treason, in impeachments the charges
could include felonies, treason and misdemeanours. The case directly
came before the House of Lords, rather than being referred to it by a
writ of certiorari. The Lord High Steward presided only if a peer was
charged with high treason; otherwise the Lord Chancellor presided.
Other procedures in trials of impeachment were similar, however, to
trials before the House of Lords: at the conclusion of the trial, the
spiritual peers withdrew, and the temporal Lords gave their votes on
their honour. The last impeachment was that of Henry Dundas, 1st
Viscount Melville, in 1806 for misappropriating public money (he was
acquitted). Since then, impeachment has become an obsolete procedure
in the United Kingdom.) . . .
Now, civil proceedings involve arrests only when an individual
disobeys a court order. Since 1945, the privilege of freedom from
arrest in civil cases has arisen in only two cases: Stourton v
Stourton (1963) and Peden International Transport, Moss Bros, The Rowe
Veterinary Group and Barclays Bank plc v Lord Mancroft (1989). In the
latter most recent case, the trial judge considered the privilege
obsolete and inapplicable, and said in proceedings, "the privilege did
not apply—indeed ... it is unthinkable in modern times that, in
circumstances such as they are in this case, it should". . . .
each peer is commonly considered a counsellor of the Sovereign, and,
according to Sir William Blackstone in 1765, "it is usually looked
upon to be the right of each particular peer of the realm, to demand
an audience of the King, and to lay before him, with decency and
respect, such matters as he shall judge of importance to the public
weal."
From Wikipedia.
Privileges Of Particular Aristocratic Positions
Also, every aristocratic office, e.g. a hereditary dukeship, originally carried with it privileges and obligations particular to the office, frequently, but not always, trivial ones, like the right and obligation to blow the first horn announcing a coronation, or the right and obligation to supply flowers for a royal garden. This has declined in importance, however. From the same source:
Individual privileges that did exist have fallen into disuse—for
example the Lord of the Manor of Worksop (which is not a peerage) was
extended the privilege and duty of attending the coronation of the
British monarch until 1937, but the right was not exercised at the
coronation of Queen Elizabeth II in 1953 as the manor was under
corporate ownership at the time.
Other Residual Privileges And Limitations of Aristocrats
As noted in another answer, a British peer may not serve in the House of Commons as a member of parliament.
In some countries, British aristocrats would be given special privileges or treatment in diplomatic, and might be treated differently in immigration matters.
Sometimes this is a positive point, and sometimes it is negative.
A British aristocrat might secure one a better place in a receiving line or table setting at a state dinner than a British commoner would.
But, a hereditary title might also disqualify you from serving as an individual in a U.S. government post that doesn't itself require U.S. citizenship.
For example, a British baron could not serve in the U.S. military without renouncing his peerage, even though you do not have to be a U.S. citizen to serve in the U.S. military, because a peerage implies a duty of loyalty greater than that of a foreign commoner to a foreign monarch.
Wealth and Education
As other answers have noted, a noble title does not automatically imply great wealth. But, in practice, British aristocrats are indeed, on average, quite wealthy.
As a practical matter, status as a peer or heir to a peerage is a plus in getting admitted to high status educational institutions, in being considered positively as a prospective spouse (especially of another aristocrat foreign or domestic), and in being admitted to prestigious clubs and business firms, although it is rarely the sole outcome determinant in any of these matters. This is somewhat comparable and analogous to the advantages someone in the U.S. would have as a legacy or graduate from an Ivy League college, or as a child of a famous or influential person (e.g. the child of a former President), in general.
For example, as of July 19, 2019:
On average, Britain’s 600 or so aristocratic families are now as
wealthy as their Victorian forebears at the height of Britain’s
imperial expansion.
The ten largest aristocratic personal fortunes left in the last decade
add up to £1.06bn when adjusted to reflect current purchasing power.
At the head of the league of aristocratic wealth stands the
high-profile figure of Hugh Grosvenor, 7th Duke of Westminster, who
will eventually inherit the personal fortune worth some £659m in 2019
prices left by his father following his death in 2016.
The 28-year-old’s total family wealth, held in separate trust funds,
is put at some £9bn.
This lengthy report goes on to examine in detail of study of probate estates of British aristocrats, suggesting an average probate estate of
£16.1m, compared to an average probate estate in 2019 dollars of £4.2m between 1978 and 1987. "From a pre-war high of £23m, average fortunes fell to £4.9m by 1967 before rocketing again by the 1980s." Elite educations and wealth have been the norm and the higher the title the more successful Britain's aristocrats have been, as the same source explains:
Of the ten largest probates between 2008 and 2018, seven of the
deceased attended Eton or Harrow, with the remaining three also
attending major public schools. Six of the ten went to either Oxford
or Cambridge universities.
Dukes, which represent the highest and often oldest rank of the
aristocracy, are also the wealthiest, leaving an average
inflation-adjusted fortune of £48m since 1958.
The next echelons of marquesses, earls and viscounts were worth £14.5m
and £11.2m, while the far more numerous barons (totalling more than
800 title holders, including the father of former Chancellor George
Osborne) were worth on average £9.3m.
Note that in Britain, confusingly, the term "public school" means an elite private educational institution, and not a government run educational institution:
A public school in England and Wales is a fee-charging endowed school
originally for older boys that was "public" in the sense of being open
to pupils irrespective of locality, denomination or paternal trade or
profession. The term was formalised by the Public Schools Act 1868,
which put into law most recommendations of the 1864 Clarendon Report.
Nine prestigious schools were investigated by Clarendon, and seven
subsequently reformed by the Act: Eton, Harrow, Shrewsbury,
Winchester, Rugby, Westminster, and Charterhouse.
These were boys' boarding schools, but some are now mixed and some
accept day pupils as well as boarders. By the 1930s, the term "public
school" applied to at least twenty-four schools, although the term has
been in use since at least the 18th century.
Public schools have had a strong association with the ruling classes.
Historically, the sons of officers and senior administrators of the
British Empire were educated in England while their fathers were on
overseas postings. In 2019, two thirds of Cabinet Ministers had been
educated at such fee-charging schools, although a slim majority of
cabinet ministers since 1964 were educated at state schools.
These wealth estimates, moreover, are minimums. As the July 19, 2019 article explains:
The figures, based on the settled estates or probates of 1,706 members
of the nobility dating back to 1858, also only tell part of the story
of aristocratic wealth by showing the minimal personal wealth of title
holders. The total fortune of the often secretive elite is likely to
be far higher because in many cases other family wealth – including
land, property and assets such as art collections or investment
portfolios – is held in separate trusts which are not open to
scrutiny.
Traditionally, however, British aristocrats held most of their wealth in land, and that element of their wealth has dramatically declined, along with their political power and legal privileges, as the New York Times recounted in a 1990 article:
In 1880 the 29 greatest British landowners possessed enormous estates.
They all had titles; 12 of them were dukes. The smallest holding in
acres -- 19,749 -- was that of the Duke of Westminster, but since a
sizable chunk of it was in London he had the biggest income of all --
$:290,000 -- just beating the Duke of Buccleuch, who had 460,108
acres, much of them in Scotland, and the Duke of Bedford, another
London landowner, who had 86,335 acres all told. Fourteen of these
landowners owned more than 100,000 acres each. The Duke of Sutherland,
whose holdings were largely in the Scottish Highlands, had well over a
million.
By 1976, the latest date for which figures are available (and they are
limited to England and Wales), the picture had drastically changed.
Although there were a few maverick cases of an actual increase, more
typical were the Duke of Devonshire, down from 133,000 acres to
56,000, and the Duke of Norfolk, from 40,000 to 25,000. Many falls
were far greater.
Of course, this did not mean that some of the surviving grandees were
other than extremely rich. The postwar boom in agricultural land and
the soaring price of objets d'art have recouped the fortunes of many
landowners who can now get by with far fewer acres. . . .
But, although the Duke of Devonshire is still one of the richest men
in England and although there is still great wealth among the
aristocracy, Mr. Cannadine's basic theme is correct. The landed world
is not what it was.
There are still supposed to be some 2,000 landed estates in Britain,
but whereas they covered over half the land in 1880, they covered at
most a quarter, perhaps a lot less, in 1980. And those who have gone
by the board are not so much the seriously rich, though some of them
have, but the country squirearchy, the men of 1,000 to 10,000 acres
who propped the whole system up, constituted the governing class in
the counties, went to Parliament as perennial backbenchers, lived on
their estates, hunted and held High Tory views. They really have
largely vanished. . . .
In 1880 the House of Lords was overwhelmingly landed and hereditary;
it had virtually coequal powers with the Commons, most of the members
of which were sons or collaterals of the Lords; and it dominated the
political scene. A century later it had become in essence a senate of
lifetime members, though the largely absentee hereditary peers were
still a majority. It is a useful revising chamber but little more. In
the Commons the old "landed interest" is negligible now. There has
never been a complete sweep of the aristocracy as in parts of Europe
but a merger with an upper middle class of professionals, financiers
and businessmen -- a new Establishment.
Still, a September 7, 2017 account makes clear that this decline in landholdings after hitting bottom, has recovered somewhat:
For all the tales of noble poverty and leaking ancestral homes, the
private wealth of Britain’s aristocracy remains phenomenal. According
to a 2010 report for Country Life, a third of Britain’s land still
belongs to the aristocracy. Notwithstanding the extinction of some
titles and the sales of land early in the 20th century, the lists of
major aristocratic landowners in 1872 and in 2001 remain remarkably
similar. Some of the oldest families have survived in the rudest
financial health. In one analysis, the aristocratic descendants of the
Plantagenet kings were worth £4bn in 2001, owning 700,000 acres, and
42 of them were members of the Lords up to 1999, including the dukes
of Northumberland, Bedford, Beaufort and Norfolk.
The figures for Scotland are even more striking. Nearly half the land
is in the hands of 432 private individuals and companies. More than a
quarter of all Scottish estates of more than 5,000 acres are held by a
list of aristocratic families. In total they hold some 2.24m acres,
largely in the Lowlands. . . .
One legal provision unique to England and Wales has been of particular
importance to these aristocratic landlords: over the centuries they
built many millions of houses, mansion blocks and flats, which they
sold on a leasehold rather than freehold basis. This meant that
purchasers are not buying the property outright, but merely a
time-limited interest in it, so even the “owners” of
multimillion-pound residences have to pay ground rent to the owner of
the freehold, to whom the property reverts when their leases (which in
some areas of central London are for no more than 35 years) run out.
This is unearned income par excellence.
Built property aside, land ownership itself is still the source of
exorbitant wealth, as agricultural land has increased in value.
According to the 2016 Sunday Times Rich List, 30 peers are each worth
£100m or more.
Socio-economic distinctions between those who were rich in 1800 in the U.K. and those who were poor then remained quite significant statistically in their descendants as of 2011 although they have been diluted significantly over the last two centuries:
averaged parent-child correlations of variables such as wealth,
education and occupation are in the 0.7 -- 0.8 range over the last 200
years, the same as found in India, with its caste system! . . .
Mobility rates are lower than conventionally estimated. There is
considerable persistence of status, even after 200 years. But there is
convergence with each generation. The 1800 underclass has already
attained mediocrity. And the 1800 upper class will eventually dissolve
into the mass of society, though perhaps not for another 300 years, or
longer.