In the UK the executive (the Government) is appointed by the leader of the party that wins a General Election, with appointees chosen from MPs and Lords (I think). But is this merely a convention as opposed to a strict rule?
If so, what would be the alternative, and has it ever occurred?
This is considered a constitutional convention, in that it's not an official, written-down rule, but it is generally expeted to be followed.
The convention that the government is seen to be accountable to House of Commons started in the 1700s, with the government of Sir Robert Walpole. In 1742, Sir Robert lost what he considered to be a vote of confidence in the government, and so he tendered his resignation to the king.
Since then, the convention has been that HM Government must always have the confidence of the House of Commons in order to govern. When a majority of MPs are from a single party, as is usually the case, such confidence is relatively straightforward for the leader of this party to receive.
When there is a hung parliament, when no party has an overall majority, things get a bit trickier. In this case, the Prime Minister will be whichever leader can get the support of enough parties to get a majority in the Commons. In 2010 and 2017, these both happened to also be the leaders of the largest parties (David Cameron and Theresa May, respectively). However, in 1923, Ramsay MacDonald became Prime Minister despite Labour being 67 seats behind the Conservatives, as the Liberals (with 158 seats) decided to support a Labour rather than a Conservative government.
To summarize: The Prime Minister is the person who can get the support of a majority in the House of Commons, and this is usually, though not necessarily, the leader of the largest party.
Almost everything in British constitutional politics is a convention, rather than a written rule, and this is no exception.
The Queen always invites the leader of the party with the largest number of MPs to form a government first.
By convention the Queen will invite the Leader of the party winning the General Election to try to form a Government, in the event of a hung Parliament she will wait until she is advised who she should invite (typically the Leader of the biggest party).
That person then has complete freedom to appoint whoever they want to fill out the Executive. Traditionally it is drawn from the ranks of the MPs and Peers but this is simply a convention, however there is a practical side in that there needs to be someone to represent the Government in Parliamentary Debates, Departmental Questions etc so the vast majority of the Executive will be drawn from the Lords and Commons.
It is correct to say this is only a convention and parliament appoints whoever it wishes to be prime minister and form a government. This is typically the leader of the largest party as they command the most votes. If the leader of the largest party can not pass the vote of confidence then other MPs may try. If parliament is unable to pass a vote of confidence to select a prime minister it would have to be dissolved and a general election called.
There is one interesting example (that I could think of) of a prime minister being appointed who was not at the time leader of a party, this was Winston Churchill who was appointed by Parliament after the resignation (from the post of prime minister) of Neville Chamberlain. Churchill formed what was called the wartime coalition, before becoming leader of the Conservative party later that year when Mr Chamberlain resigned from the House of Commons for health reasons.
Perhaps most interestingly Wikipedia suggests
Churchill probably could not have won a majority in any of the political parties
