Bevan would have made a great Prime Minister imo
I'd imagine the earliest possible period for Bevan to consider taking over would have to be after the Korean War. So, say Attlee resigns after 8 years at the end of 1953 to give a full year or more run up to an election and Bevan takes over. Bevan's premiership manages to bring in fresh blood as many of the older cabinet members retire, etc. Meanwhile, Churchill accepts defeat and resigns around the time Bevan takes over, with Eden succeeding him. The 1955 election is roughly a repeat of '50 albeit the Labour vote shrinks a bit yet they end up gaining a few marginals.
Due to this, Eden and Macmillan come to an agreement to revive an idea of Churchill and Macmillan's to merge with the Liberals and form a New Democratic Party. Talks go through and an initial agreement ends up being something like that of Australia's National and Liberal Parties. Eventually they fully merge into the NDP.
In the meantime, Suez occurs with Bevan dealing with it in a more diplomatic way though ultimately failing to make any real difference to OTL. However, Eden still pushes for invasion etc and his view is much more popular amongst the British public with Eden soaring in the polls over Bevan. However, aside from local election wins, etc, this starts to fizzle away yet Eden keeps a constant single-figure lead.
In the late 1950s, revelations start to emerge over Gaitskell's affair with Ann Fleming. However, these reports are largely sidelined after Eden's stomach complaints worsen leading to his death from an accidental overdose of Benzedrine. Bevan resigns in 1959, with some speculation over his health. Gaitskell makes a shock victory promising the 'fight and fight to save the country that he loves' from a return to 'the Emperor in New Clothes' (Macmillan, a reference to the party's change of image in the NDP).
Gaitskell short period in office is quite tumultuous due to wrangling over the 1960 manifesto, the worsening of Britain's economic situation and the oppositions criticism of Labour in the 1950s citing the 'white elephant' of the Festival of Britain's use.
By the 1960 election, the NDP win a sizeable victory gaining a 93 seat majority. Gaitskell is largely unpopular after private life revelations return to the fore whilst Macmillan attacked their economic record despite consensus meaning they'd have done largely the same. Macmillan's 3 years in office would be marked with the boom in satire, particularly Macmillan's change in remarks from 'never had it so good' before the economic downturn compared to afterwards. Macmillan also has to stymie the remaining Liberal contingent pushing for proportional representation and more progressive policies and lower taxation. Macmillan is labelled a traitor.
Macmillan's internal party struggles continue whilst they suffer their own crisis in the Profumo Affair followed by Macmillan's illness in 1963. Owing to new party rules agreed by the merger, the NDP members would be able to vote for the new party leader. The result was a Butler victory. Passed over numerous times, Butler would now be able to lead his party... and now his country.
Forced to juggle economic difficulties and prominent rising stars such as Edward Heath, Jo Grimond, Enoch Powell and others, Butler would have a difficult task. It had appeared certain that Labour would win the next election. However, following the death of Hugh Gaitskell, the situation suddenly changed. Labour had remained united behind Gaitskell, albeit precariously, throughout Macmillan's period in office, but after his death tribalism emerged. Labour would split into three distinctive groups behind Wilson, Brown, Callaghan and Castle.
Many in Labour blamed Gaitskell for the loss, others blamed the policies pursued under Bevan which possibly caused the economic difficulties. The result was a surprise victory for Brown with many preferring to continue with the 'devil we know'. Others have proposed Brown was chosen through conspiracy, knowing that they'd lose in 1965 anyway, due to the difficulties of overturning such a large majority.
By the time of the election, this was largely true, Brown managed to regain a number of seats cutting the government's majority to 39. Butler's second term would again be plagued with strikes and economic problems. Yet figures such as Enoch Powell pushed for reforms and new ideas in agreement with the old Liberals. Thus, a number of social reforms made their way onto the agenda such as decriminalisation of homosexuality, divorce reform, and the abolition of theatre censorship.
However, the Butler Government is remembered mostly for the involvement of Britain in Vietnam in return for economic assistance from the United States. Thus, Britain was able to prevent a devaluation of the pound and invest in economic recovery thanks to Johnson's 'blank cheque'. Whilst this was pragmatic in economic terms it was hugely criticised inside and outside the party. Yet, tax policies were still criticised by Powell and The Beatles (Taxman in 1966 remarking 'ah-ah Mr Butler, ah-ah Mr Brown')
Meanwhile, the drunken Brown was given the boot, with a surprise victory for Barbara Castle. Castle, as a woman, gained huge support especially amongst youths and in popular culture. Castle criticised Butler's administration highly successfully for its employee relations due to increasing strikes, its foreign policy due to the wars and more, meanwhile, her deputy in Harold Wilson focussed on policy and PR, proposing new acts based on ideas lower down the Labour ladder such as an 'Open University' and a 'technological revolution'.
By 1970, the economy had recovered to an extent yet Labour made much political capital out of the collapse and nationalisation of British Motor Corporation, the waste of time attempting to gain entry into the EEC and the Vietnam War. Castle proposed a 'road to removal' from Vietnam, an Open University, more economic investment and the pass of 'In Place of Strife' to help prevent militant trade unionism. Labour would win a majority of 61 seats.
The NDP would descend into chaos with various factions warring with each other. Many remarked it was the election that destroyed the Conservatives, others saw it as a victory for the Conservative left following the departure of the party's 'right', figures such as Enoch Powell and later Margaret Thatcher and Keith Joseph would abandon the party shortly before the 1974 election forming the New Conservative Party.
I'll stop there. Got carried away.
1945-1950 Attlee (Labour)
1950-1953 Attlee (Labour)
1953-1955 Bevan (Labour)
1955-1959 Bevan (Labour)
1959-1960 Gaitskell (Labour)
1960-1963 Macmillan (NDP)
1963-1965 Butler (NDP)
1965-1970 Butler (NDP)
1970-1974 Castle (Labour)
1974-1977 Castle (Labour)
1977-1979 Callaghan (Labour)
1979-1983 Heath (NDP)
1983-1985 Heath (NDP)
1985-1988 Heseltine (NDP)
1988-1992 Owen (Labour)
1992-1996 Owen (Labour)
1996-2001 Blair (NDP)
2001-2005 Brown (Labour)
2005-2010 Brown (Labour)
2010-2015 Clarke (NDP)