[throwaway4aday]

It's more likely that a nuclear exchange would start with limited use of tactical nuclear weapons. Let's imagine that Russia does invade a NATO country and is beaten back by NATO forces after significant fighting within NATO territory. Maybe some of those hawks higher up the chain of command want to make sure Russia can't regroup and try again so they pursue Russian forces into Russia itself in an attempt to encircle them and force a surrender of a significant number of their troops. Russian higher ups misinterpret this as a push to capture Russian territory or the start of a larger invasion and decide to use a small number of low yield nuclear weapons to destroy the invading forces. Now you have thousands of dead NATO soldiers and a mess of confusion with multiple nuclear detonations on the Russian border. NATO counterattacks by targeting airbases and missile assets using cruise missiles and aircraft with conventional warheads in an attempt to prevent any further strikes. At that point things can heat up quickly.

In a hot war where multiple countries are involved you can't expect the fighting to just stop once you hit a border. Even when you have the upper hand, the temptation to carry through once you are already engaged in a conflict is real.