Can we keep it to a proxy war? America/UK/France, the USSR, and Germany never...quite...declare war on each other. The Soviet Union invades Finland, and the Allies declare war on the Finnish People's Republic, allying with the rump Finnish government. Soviet troops fight American, but not under those terms. Poland is a simliar situation, as Soviet troops fight Germans, under the guise of the Polish SSR. Britain invades Norway and fights Hitler, but the declaration is against Hitler Norweigian proxy. Something like this, carried out in the colonies, etc. to an absurd degree, with alliances between the three powers springing up by theater?