In the UK or other countries that follow Westminster First Past the Post style elections, what cases have there been where the Prime Minister is not the leader of the largest party, (discounting explicitly interim holdings to cope with death/resignation/etc and frictional time for the Monarch/Governor General/etc to give consent)?
For the UK, a few coalition governments qualify:
The 1918 election left the Conservatives with 382 seats, giving them a majority. However they decided to continue with the wartime coalition with Lloyd George a Liberal Party member (and George didn’t even lead that party).
In 1931, Ramsay MacDonald formed a coalition with the Conservatives and Liberals. A mere 12 members of MacDonald’s Labour Party joined him and he was even expelled from the party. Somehow this state of affairs lasted until 1935, when Conservative leader Stanley Baldwin took over as PM.
When Chamberlin gave way to Churchill in 1940, Chamberlin continued as Conservative Party leader for several months. When he resigned due to illness later in the year, Churchill was chosen as party leader.
In 1923, the Conservatives, led by Baldwin, won the most seats in Parliament, but Labour, led by Ramsay MacDonald, and and the Liberal Party had more total seats and McDonald became Prime Minister.
In the Canadian federal election of 1921, Prime Minister Mackenzie King and the Liberals came in second in the seat count (100 seats), behind Arthur Meighen and the Conservatives (115 seats).
King was able to form a minority government with support from the loosely organised Progressives, who had 22 seats. (They disliked Meighen and his policies more than they disliked King and his policies, particularly the tariff issue.)
Ontario 1985: the Progressive Conservatives came in first in seats, but were defeated in a confidence motion by the Liberals and NDP. The Liberals formed a minority government.
Quebec 1878: Henri-Gustav Joly de Lotbinière and the Liberals won one less seat than the Conservatives, but formed a minority government with the support of two independent Conservatives.
British Columbia, 2017: Premier Christy Clark and the Liberals won 43 seats, while the NDP under Horgan won 41 seats. The Greens won 3 seats.
Horgan reached an agreement with the Greens and is currently the Premier.
(Clark advised the Lt Gov to call new elections but the Lt Gov rejected that advice and called on Horgan to form a government.)
Saskatchewan 1929: Premier Jimmy Gardiner and the Liberals won 28 seats. Anderson and the Conservatives won 24 and the Progressives won 5.
The Conservatives and Progressives defeated Gardiner’s government in a confidence vote in the Assembly and formed a coalition government.
Just as an aside: curious why the OP restricted the inquiry to FPTP systems? Same situation can arise in parliaments with other voting systems.
Not the Westminster system but something similar happened in the US House of Representatives in 1854 and again in 1916.
Ireland, 1948 and again in 1954; there was a multi-party government, which (a) did not include the largest party in the Parliament (that party formed the opposition), and (b) was led by a Taoiseach who was a member of, but not the leader of, the largest party in the governing coalition. Not a first-past-the-post electoral system as specified in the OP, though but, like Northern Piper, I don’t see why FPTP was specified.