The challenge here is to have something analogous to the Cold War take place, but to have 3 or more sides that are all working against each of their opponents. It can't be something like "the USA is not a fan of the British empire", it has to be full on opposition from each side to each other side. Bonus points for extra sides.
Have national Socialism aka The Nazi's survive past WW2
Perhaps have Britain and France Rearm more effectively earlier and be powerful enough to force a stalemate in 1940 or the equivalent year for this TL but not be powerful enough or have the desire for a an expensive costly fight.
Russia rebuilds her armed forces before Germany invades (any OTL type of Barbarossa) is delayed due to the western campaign bogging down and need to keep forces facing he French and British and for her Part Russia is distracted by internal issues (yes I'm making this up as I type) so does not get around to
So you end up with
Axis powers - Germany, Italy, Spain, Nationalist China
Communist powers - USSR, those Eastern European Nations that have 'joined' them + Communist China
'Western' powers - British Commonwealth, USA, French Empire and Japan (?)