My pessimistic take is that it is currently about 2%, or 1-in-50, which is far too high for comfort.
The US is confined by its Wolfowitz Doctrine, Russia is confined by its history of 25 million dead in WW2.
Neither is capable of backing down. Things will proceed to nuclear.
Also, "things will proceed to nuclear" means that you think Putin will initiate the end of human (and most plant) life on Earth.
I don't think the richest man alive, who seems to desire material possessions and power over almost anything else, would want the world to end completely. He could leave office now and have an incredibly pleasant life.
Actually, I think the initiation will come from the US side. Read up on the Wolfowitz Doctrine. The US will not allow that they lose the 'Number One', 'Exceptional' crown and will do anything to prevent that.
Joe Biden is not going to initiate nuclear strikes. Russia has automated nuclear weapons that would launch without human intervention, leading to the end of the world.
If Biden doesn't initiate nuclear strikes and Russia wins this war, the only thing lost would be Ukraine.
If Biden does initiate unclear strikes, the entire world would be lost.
Biden is not going to sacrifice all human life to save Ukraine.
Read up on mutually-assured destruction.
I hate to break this to you but the neocons are still well in control of the US. Wolfowitz is a neocon.
Nuclear weapons are their most powerful when not used.
More seriously, you have an un-hinged dictator with nuclear weapons - when he loses, who knows what he'll do. A real egotarian would say "If I can't win, then everyone has to lose". The only upside is that there's one less dictator (wannabe?) with his finger on the nuclear button than there was last year.
Talking about Russia, AFAIK only Putin that is cornered enough to want to use it (and his aides I guess). The world, AFAIK, signals that they won't have problem forgiving Russia if the current regime changes. So I guess it's higher for civil war / coup in Russia before nuclear solution.
I don't know with smaller tactical nuclear though (district size maybe).