In the later days of of the fighting in France in 1940 plans/proposals for a Franco-British were drawn up as a way to keep France fighting after the loss of the French Metropol. However after the British accepted the French request to seek a separate peace the proposal was no longer relevant to the war. The proposal involved the uniting the two nations into one for the period of the war, with joint citizenship, joint trade, a joint currency and joint military command. In the case of an axis victory in the east what would the fate of the war and the union be, or how would it carry on. I'll briefly lay out a scenario that seems to me plausible at first glance, as prompt if needed.
June 40:
Franco-British union agreed, and the evacuation of French government to London or first Algeria then London. Some puppet government established in German occupied France. French colonial empire remains at war with Germany.
September 40: Japanese do not occupy Indochina, resulting in no US oil embargo that year.
Winter 40-41: Italian Libya conquered by Franco-British forces.
Spring/Summer 41: Axis invasion of the Soviet Union. Japanese invasion of the Russian far east. US oil embargo on the Japanese
Autumn/Winter 41: Vast territory seized by European axis forces including the fall of the Soviet capital in Moscow. Comparatively small gains by the Japanese in the east. Capture of Vladivostok and North Sakhalin.
42: German victory in and occupation the Caucuses. Fall of Leningrad, Finnish occupation of Murmansk. Substantial reduction of the Red Army's offensive capabilities and Logistical issues involved with further German advances are enough to prompt peace talks between Moscow, Berlin and Tokyo.
43: Peace signed between axis and the Soviet Union. Japanese declaration of war on the Western allies and perhaps the US as well.