World War III probably. But remember at this point in time the USSR's nuclear capabilities are very limited compared to the USA. They don't have good delivery systems and was outnumbered by US nuclear weapons by something like 10 to 1. If a nuclear exchange occurred it would be very one-sided. The USSR might be able to take out some major centres - Berlin, Paris, Seoul, Tokyo, etc and they might use nukes against US ground forces but the US would be able to achieve much more than that. Beijing, Shanghai, Tianjin, Lushun and Guangzhou would be reduced to dust, and most likely Moscow and Leningrad as well.
It's an interesting scenario. Fighting would be bad in Europe and China. The Chinese communist government might collapse, allowing the Nationalists an opportunity to return to power. The Soviet Union might survive, but in a very bad shape. America would probably be victorious. If not, then it'll at least be better off than the communists.