That was not a nuclear war. If one country gets nuked today, the rest of the world nuclear arsenal will be expended.
There is an infinitesimal chance that only one nuke goes off, or only one country nukes and the other doesn't.
This is why Mutual Assured Destruction is such an effective peace time strategy, until it isn't.
The sad thing is - the madman holding the nuke button has all the control. He's crazy so he'll do anything. He's old enough not to care about dying and destroying the whole world with him.
Everything doesn't just swing into action automatically. Poland gets nuked. Congresses, UN, Nato all hold virtual summits to figure out what to do. A week later, they decide this can't stand but if we can save 5 million lives in NYC, maybe we should.
It would like play out more like chess by mail, Putin makes a move, Biden makes a move, maybe Germany makes their own move, etc.
Once the game is in motion, it can go many ways. All Russia needs to do is cut a deal with a few NATO countries to let them be 'neutral' if they keep their noses out of things, and Nato falls apart, and then it leaves USA vs them, of course the way Russia military runs things that's not a big threat.
Another thing to think about - most of Russia's military officers are corrupt. They've been embezzling funds meant for supplies/equipment etc for years and now they finally need that equipment, they're getting shown off for fraudsters.
It costs iirc something like a billion every 10 years PER nuke just to update them and keep them working. On top of that I've read there could be as high as a 60% failure rate for normal nukes in action, probably due to the fact they mostly spend their entire lives just sitting in a bunker somewhere. Quality checking may not get done as often because of the radiological risks etc.
I'm betting of all the nukes in existence, Russia's by a magnitude of 20 maybe are more likely to fail than China or USA or other countries that actually have a competent military.
Furthermore, however insane any leader of a nuclear power may be, I just got to believe there are competent, clear-headed people in the chain of command who would refuse to follow suicidal orders while not under immediate attack themselves.