"Alas, Babylon" was published in 1959 and was written by Pat Frank. It has been many years since I read it, but if I recall the problem began when a heat seeking missile hit the wrong target in a Middle Eastern locale known as Latokia. Through the cold war tensions of the time the superpowers were dragged into it. The Soviets launched. The U S counter launched, and the "fun" began.

The story deals with a post war devastated America. It focuses on a small radiation free pocket in Florida, and deals with the lives of the characters trying to live on increasing shortages of the necessities of life and the knowledge that it is simply a matter of time until they are also wiped out by the approaching radiation.