The game will have three stages, and the invaders will follow those stages:
1 - The period from the start of the nuclear war (very end of Dr. Strangelove) and until the mine shaft is closed off to the outside world. Will be around 3 months, up until the atmosphere has been completely saturated with radiation and your mine shaft has to be closed off. During this time period, Communist forces will probably make an attempt or two to sabotage or right out destroy your mine shaft project. One of the things about the end of Dr. Strangelove is that the Cold War parties end their race for nuclear armament and begin a race for getting the most mine shafts operational, avoiding a mine shaft gap, so that one side can resurface stronger than the other when the planet is habitable again. So the Communists will try to either destroy or conquer your mine shaft in phase 1.
2 - The holding out period, around 100-200 years. Your mine shaft is completely cut off from the outside world and all forces above ground will slowly die. So this period is more about internal security. An important part of the paranoia of Dr. Strangelove is the fear of saboteurs, and in the game, at the very least 1 of your units will be a Communist infiltrator. Always. Even if only two people are left, one of them will be an infiltrator. You, the player, won't know who it is. But the Communist infiltrator will perform acts of sabotage, murder, espionage, and he (or she) will also recruit more infiltrators. If they become strong enough, the mine shaft will be thrown into civil war.
3 - Radiation levels have fallen to an acceptable level and the outside world can be accessed again. I'm thinking we are moving into a more Fallout-like setting now, where each surviving mine shaft settlement fights for control over resources and perhaps some survivors have held out on the surface and will want to get your mine shafts technology and so on. Maybe the US and Soviet governments still exists, maybe they don't. Will probably be different from game to game. This last phase will probably feel more like a classical DF game, with an underground and above ground environment.