France fought beside the Americans and Brits for the duration of the war. I can’t see them siding with the Soviets; at least not right away (perhaps the USSR eventually wins massive swaths of Europe, forcing the French, Romanians, and Polish to fight for them). If war broke out I don’t think the UK would stay as an intermediary for very long. Once France fell to the Soviets they’d see the Red Army as a threat.
The Free West: British, Australians, Canadians, Americans, Free French (DeGaulle quotes as sighing, “Here we so again!”)
The Red Army: Soviets, French, Polish, Romanians, Conscripted Germans/Looted German tanks
One of my favorite video games has alot to do with this. Freedom Fighters assumes that the Soviets actually dropped the first atomic bomb, on Berlin. Thus began the assimilation of all of Europe. Eventually the war reaches American shores.