Well, those troops can't fight in Europe if they're radioactive dust either. Both sides will have expended all of their Straegic missiles, ICBM, SLBM, IRBM, whatever within the first hours. Then it's up to the bombers. The US bombers will get to the USSR within two to three hours from their forward bases. I don't know how long it would take Soviet bombers to reach the US, but it's probably longer.
On the ground in Europe, the Soviets have launched their strategic nukes and probably taken out a good number of European cities and forces. At this point, since it's already nuclear, Neither side has qualms about using tactical warheads to stop the enemy. Also, considering the weight of the Red Army, it's the only chance to keep Western Europe from being overrun.
Even after expending tactical nukes to slow the Soviets, the US still must keep the sealanes open, as there are going to be more Soviet forces on the way, and the only choices are to either irradiate Europe to the point of being uninhabitable, or to land conventional forces and oppose them on the ground.
NATO would probably push for conventional forces, and the USN, which still has an edge on the Soviet Navy, would have to destroy the Soviet fleet and keep the convoys safe from attack.
So, two possibilities
1) Severely Radioactive Europe, USSR, Cuba, and to a lesser extent US.
2) Severely Radioactive Germany, USSR, Cuba, and to a lesser extent Western Europe and the US, with Western Europe either in the hands of NATO or the USSR.