“Nuclear winter”
A full scale war would involve thousands of nuclear devices, each one being perhaps hundreds of times more powerful than what were used in World War II. There is no way to model what would happen but the realization that the losing side could simply fire all its weapons and destroy the world (and in effect, prevent the winner from occupying the loser’s territory) was what made “mutually assured destruction” so effective.
There other ways to induce such a “winter”, weather as a means of deliberate destruction, or a misguided attempt to reverse global warming. Please note that in the past, “warm” has generally been good (probably since the longer growing season is good for food supply).