As used in these stories. A war that is mainly fought with the use of fissionable and radioactive materials.
Nuclear detonations occur when fissionable material is compressed and heated to thermonuclear temperatures of at least 100 million Kelvin degrees.
The only global nuclear war that ever happened destroyed most of the animal life on the surface of our world as a result of the nuclear winter it called forth. We call that war and its aftermath the Catastrophe.