A very well conceived TL about this scenario, upon which people can get some inspiration on the effects and possible evolutions is in Amerigo Vespucci's
The Cuban Missile War Timeline.

I also added in that thread some basic political info about the world as of October 1962, if anyone needs it.
That's actually a pretty good scenario for 1962. A nuclear war at that time would be more limited than in the years of MAD and the US would have clear superiority. The scenario could be a little less severe:
1. The ICBM's would be deployed as described. But the Soviet bombers headed for North America (Ohio, N. Dakota, Chicago, Ottawa) might be shot down by US and Canadian forces. Likewise, NATO bases in Europe might survive longer for the same reason, increasing the damages to the Soviets.
2. The Soviets would direct their arsenal to NATO, so the southern hemisphere would be very much spared.
3. The notion of President Johnson being deposed and sent to Leavenworth is not realistic and not necessary for the war timeline. The remaining military would not let it happen. There would be a regular election in 1964.
4. The choice of St. Louis as a capital makes sense because (a) with Colorado Springs and Topeka nuked, the designated sites of Denver and Kansas City would be less attractive; and (b) in the spirit of north/south politics, the site lies right next to the "Land of Lincoln" but in a former slave state.
5. The loss of the Detroit auto industry might have less impact than you would think. The US went more than four years without new cars in WWII, and assembly plants and suppliers could be rebuilt faster. There would be spared manufacturing capacity remaining in the US, Canada, Asia and the southern hemisphere.