The Demon Drink..
I can see only two ways Brown can win the leadership - either by being sober and sensible or by butterflying away Wilson or having an equally strong candidate on the Left.
Brown, apart from his personal embarrassments (the events of 22 November 1963 being a good example) was politically out-manoeuvred consistently by Wilson, arguably one of the shrewdest political operators of the post-war period.
Brown's drinking was well known in Labour and Westminster and there were those who would otherwise have supported him as the standard bearer of the Gaitskellite wing who were appalled by the prospect of him being a potential Prime Minister.
A sober and sensible Brown would have won the Labour leadership and probably won the 1964 GE by a larger margin than Wilson. He would have had the problem of what to do with Wilson and Callaghan - perhaps Wilson as Chancellor, Callaghan at the Home Office and maybe Jenkins or Crossland as no.2 to Patrick Gordon-Walker (who might have survived with a better Labour performance).
It's hard to see how Brown would have avoided the devaluation crisis to be honest - would Wilson have resigned as Callaghan did ? Seems implausible.
Could Brown win in 1970 or 1971? Not sure - if he does, the Labour Government faces the oil shock of 1973-74 and, given Brown's Jewish origins, some hard problems over Mid-East policy. I simply can't see that ending well.
If Brown is defeated in 1970 (and assuming he keeps his seat which he didn't in OTL but seems much more likely) does he stay on ? I think so with Wilson or Callaghan as his Deputy.
Beyond that, it gets very murky - IF the Conservative Government falls in 1970, Brown might get back into No.10 in 1974 and will serve out a difficult term before a second defeat in 1979 after which he steps down (Wilson has already quit politics and Callaghan is too old). Jenkins, Healey and Foot fight for the leadership which Foot wins.
Brown becomes a high-profile member of the new SDP in 1981 and is able to persuade more of his former MPs to join (perhaps up to seventy) and the SDP survives as a larger more coherent force through the 1980s.
In OTL, Brown died in 1985 at the age of 71 but no reason why, without the drink, he couldn't live longer and become the "grand old man" of the SDP keeping it in the Gaitskellite tradition (which he sees strongly in David Owen).
In 1992, he approves of the SDP's decision to support John Major's minority Conservative Government.