Supposing that Putin did launch a nuclear nuke, USA and NATO wouldn't simply ignore such a situation, just because Ukraine isn't in the EU or NATO. It would be a disastrous turn, since it would launch a series of nuking from both parties, leading to thousands if not millions of deaths, while leaving an uninhabited earth.

I do get your point, the less Putin has to lose, the more dangerous he becomes, I don't think that there are any more sanctions to implement, the condition is already a mess, he has already lost, that's what makes him threatening, he doesn't have anything else to lose, so he might as well take humanity with him.