It depend when this nuclear war would be and who would be involved and how devastated countries are after NW.
But if we suppose that nuclear war is WW3 and it not be extremely apocalyptic, these countries might be nuclear powers. Probably best case for this would be hotted Cuban Missile Crisis.
United States
Russia (if it not nuked to ground)
United Kingdom
France
China
India (if it is still willingful make nuclear weapons after such devastation)
Pakistan (same as with India)
Israel (if it survives from war and want still create such weapon.)