I don't see how the Soviets putting a man on the Moon before the US and having all of Germany somehow means they'll ever be in a place where they can invade the United States. Even if all of Europe was satellites of the USSR, they still wouldn't stand a chance in hell of invading the United States, and that's before you get to nuclear weapons.
Okay, let's go for a long-shot here. Is this really, really implausible? Yep. So let's give it a go, based on what @lwhitehead has given us.
So the Soviet Union has managed to Win the Space Race in the 60s, making practical advances in rocketry and metallurgy that took longer for them OTL. The US lost the Korean War in the 50s, and the Sino-Soviet split doesn't happen. Let's allow that Western Europe ends up drifting more pro-Soviet than they did OTL, even. Throw in the stereotypical "America turns back inwards" motif, withdrawing from socialist Europe (possibly being encouraged to leave). The movement for nuclear disarmament increases, and it slows American nuclear development.
Korolev gets the N-1 working, and sooner, before his death. Someone in the Soviet Union gets Pournelle's research in the 50s from Boeing on the idea of Rods From God, or comes up with it themselves; either way, the combination of improved rocketry and metallurgy means that by 1970, the Soviets have managed to put in numerous orbital kinetic bombardment tungsten weapons. They have the high ground.
Using the threat of unleashing kinetic weaponry, the Soviet Union, leading the "peace-loving nations of Europe and Asia", demands that "the warmongering, imperialist United States" stand down. Perhaps there's an initial resistance -- and then tungsten rods smash into some American city, destroying it. Chicago, or St. Louis, or something: something inland. A few of these, and the American government capitulates, and a Soviet-led occupation force moves into America to secure their nuclear weaponry.
This is all ridiculously improbable, but it could all be strung together, I suppose.
I've got nothing for the idea of a Secret White Russian Resistance movement, though. That seems a bit odd to me.