Incidentally, as a 'Wykehamist' (old boy of Winchester School) Gaitskell would have been the first ex-Winchester PM since Pitt the Younger's sidekick Henry Addington (PM 1801-4) rather than this role going to Rishi Sunak. Gaitskell had the same well-off middle-class background and private school education as Crosland but was as far as I've read not much interested in egalitarian education theory, unlike C, and would have backed off from any idea of legislative action against the public schools rather than clash with the 'Establishment' to which he belonged (not least as a keen socialite and London party- and club-goer, in which he was a complete contrast with Wilson). But as a former part-time Workers Education Association lecturer in the 1930s he would probably have been keen on expanding the red-brick universities and vocational training via polytechnics and other colleges to open up access to skilled jobs for more working class people , if not as committed to the Open University unless a fan of the latter who was close to him had persuaded him - and he was , unlike Wilson, a political foe of the Wilsonite arts minister and OU supporter Jennie Lee. his ex-rival Bevan's widow, who was unlikely to get a job from him.
More and earlier promotion for ex-Communist turned centrist moderate Denis Healey, Roy Jenkins as the 'Number Two' in the govt instead of the working-class ex-union official Callaghan, an earlier 'In Place Of Strife' or its equivalent with a Blair-style enthusiasm for attracting centrist voters by marginalising Trade Union influence and standing up to wildcat strikes, but HG continuing his personal opposition to the EU so no Wilsonian style second attempt to get into the EU 1968-9 and possibly a rift with Jenkins over this as 'missing the boat'? Jenkins resigns and is eclipsed as the natural successor by either Crosland or Peter Shore, and once HG either loses an election and retires as leader or retires anyway (c. 1972, after a successful second election victory and largish majority in 1967 or 1968?) a struggle between marginalised Left and dominant right in the party? Led by Foot or TU-supported Callaghan for the Left, vs either Jenkins or (if the anti-EU line in Downing Street has led to him giving up in a huff and going off to a Brussels job) Crosland/ Shore? But in this scenario, with a presumably strongly technocratic and 'proto-Social Democrat' faction in charge in the Cabinet in the late 1960s, do we get Benn the then technocratic enthusiast (postal then technology minister in Wilson's govt 1964-70) staying with the moderate wing of the party as anti-EU and challenging Crosland as a more populist, pro-TU, and youth-friendly alternative?
Would Gaitskell be more engaged with anti-USSR activism and revolts in eastern Europe esp in 1968 in Czechoslovakia, and thus closer to LBJ, than Wilson, given his earlier involvement in anti-authoritarian local resistance (to the Dolfuss regime and the Nazis) in Austria as a visitor there in 1934, and his emigre Austrian Jewish wife? And so inclined to sign up to sending troops to Vietnam as a US ally, enraging the Left and so exacerbating late 1960s splits in his party? Would this even lead to a 'Socialist' secession to form a new party with militant union funding from more pro-Moscow TU leaders in the early 1970s if he kept an iron grip on the party and cabinet promotions and was seen as a 'US stoodge' by the Left , esp younger members, cf Blair after 2003? With collapse in Vietnam as a 1970s version of the real 2000s Iraq war as a lightning rod for a demand to 'take the party back' or leave, and the party leadership having to choose between Atlanticism (under Healey?) or a pro-EU move under HG's successor?