In the aftermath of World War Two when all the major British colonies were either granted independence (India, Pakistan) or already self-governing Dominions (Australia, Canada etc) it would be quite feasible for this to be taken a step further.
Here is my small little timeline to show the lead up to the break up of the UK. The POD is that there is no Marshall Plan assistance for the UK.
1946 - With the break down of the British economy following the exhaustion suffered in World War Two, Scottish, Welsh and Irish nationalism is on the rise, with some political parties formed.
1947 - The Labour government of Atlee falls after its failure to pass the annual budget through the houses of Parliament.
1948 - Coalition government of Conservatives, Tories and the Nationalists is formed. A motion to repeal the Act of Union (1701) falls short by a handful of votes. However plans are formed for greater Scottish autonomy.
1949 - 52 Nationalist sentiment in all countries of the United Kingdom continues to build. King George VI.
1952 - Queen Elizabeth II is coronated and uses the occasion to call for greater co-operation from 'all nations of the Commonwealth' This is interperated as covert support of Scottish, Welsh and Irish aspirations.
1953 - The Scottish, Welsh and Northern Irish National Parliaments Act is passed by the Coalition government. This gives control over internal affairs to these political bodies.
1955 - All the new national parliaments pass the Act of Westminster (1932) which fully seperates these parliaments from British polical control at Westminster effectivley making them Dominions.
1956 - Act of Union dissolution Act is passed through British houses of parliament and given Queen Elizabeth's consent. She will remain the Head of State and Monarch of the new Commonwealth Realms. (along the same lines that she is the Australian or Canadian Head of State)
The Melbourne Olympics would also be the last Olympics that a UK team would attend instead of seperate teams for England, Scotland, Wales and Northern Ireland.
Of course there would be some backlash and violence from these moves, particularly in Northern Ireland. Ultimatley they would remain part of the British Commonwealth, but would be independent nations.