It's unlikely that there will be a nuclear war. I'm a lot less confident than I was just a couple of days ago, however. But still, I think the chance of a nuclear attack in the coming months is under 10%, i.e. we've got a better than 90% chance of there not being one.
I read a BBC article earlier that made the point that Putin has repeatedly shown that the phrase "he'd never do that" doesn't apply to him. That is a concern because it throws the nuclear deterrent argument out the window. Traditionally nuclear stockpiling has been based on the MAD (mutually-assured destruction) principle - I can't attack you because you will destroy me.
Putin has stated that if there is a world without Russia, there is no need for the world to exist at all. On the face of it this complies with the traditional MAD idea - someone tries to wipe Russia out, they'll retaliate. However, it is becoming increasingly clear that perhaps Putin already thinks Russia is being wiped out. His vision of Russia is a USSR-style one, not a democratic capitalist society.
No rational person would believe that NATO or the EU are a genuine threat to Russia. We're not going to invade them, if they leave us alone we'll leave them alone. The trouble is, Putin sees the former USSR territories like Ukraine, Slovakia, Estonia etc. as being part of Russia's legitimate sphere of influence. He doesn't accept that they have the right to have a genuine democracy, he doesn't accept that they have the right to join NATO or the EU. In his eyes, his vision of Russia is already under an existential threat.
In Putin, the world isn't dealing with a Trump or a Johnson - people who will lie and deceive to gain influence, people who are out to line their own pockets, but ultimately know that war costs money and lives, which doesn't help them in their aims. Putin is only interested in being remembered as a "great" leader in the same way Lenin, Stalin or Hitler are. ("Great" as in successful at making their mark on the world - not as in nice.)
If he's not remembered as a successful conqueror, there's a fear that he will see the destruction of the planet as a better alternative than being remembered as a failure.
A nuclear war is still unlikely though.
Maybe 80/20 against.