Are major northern urban centers destroyed? It's my understanding that a Cuban Missile War would leave most of the United States relatively unscathed, as the Soviet Union lacked the military capacity to hit the American core at the time (Europe is another story). But I don't know specifics.
The Civil Rights Movement as we know it was the product of large amounts of urban minorities outside of the South gaining access to education, economic pull, and voting rights before and during the post-war boom. Once this intelligensia transformed into a political leadership, capable of organizing national movements, they were able to exert pressure and expose the hypocrisy of the United States, which was trying to win over the world in the court of public opinion against the Soviet Union. If urban centers are wiped out, social progress, racial and otherwise, is going to be ground to a halt. If they aren't, then I'd expect movements to resume once the war and the aftermath has been dealt with sufficiently.