The chance of nuclear war is incredibly small. In Cold War 1 there were 5x as many nukes as now, but there was never a war.
Even Putin is scared of Mutually Assured Destruction. Russia nukes, NATO nukes back. Billions of people would die, and even Putin will not want that. It'll be a suicide mission for him. There will be no-one left to dictate to or control. It's perhaps the one thing Putin is scared of.
Obviously in the last few days things have happened that last week no-one thought was remotely likely but this situation is nowhere near as tense as Cold War 1 and no-one got nuked then. At max this will be a Cold War 2.