Nuclear war was arguably the first existential risk that humanity created. It seems to be a very clear one: a full scale nuclear war leads to human extinction.
However, we apply a strict definition of existential risk: only events leading to actual human extinction count. It is actually quite hard to make humanity go completely extinct with nuclear warfare. Cities can get bombed relatively easily, but places with low population density would be much harder to destroy. Therefore, complete extinction because of direct effects of nuclear war, although very important for non-existential reasons, is not the main existential threat.
The main route towards extinction because of nuclear war would probably be nuclear winter. In case of a nuclear war, firestorms from targeted cities would create giant columns of smoke which rise high into the stratosphere. They might remain there for about five years, blocking sunlight. This could lead to a global temperature decrease of seven degrees during those years, after which temperatures would slowly return to normal. This fall in temperature would make growing food impossible in many of the currently most productive areas, probably leading to mass hunger.
There is a possibility that nuclear war would cause extinction, but the possibility is only very slight, estimated at 0.1% for the next hundred years.