I’m not sure if using nukes would get NATO directly involved. First, it would be counterproductive wrt Putin’s goal above. Second, it could provoke strikes against NATO - Russia stands no chance militarily, but they can still kill civilians. Third, I don’t think isn’t even necessary - looking at experiences so far, Ukraine is more than capable of wiping the floor with Russian army, provided they get the hardware necessary. So far NATO has been very (way too much imho) careful to only provide Ukraine with defensive weapons; they didn’t give them cruise missiles or fighter jets, for example. Likewise Ukraine so far avoided striking anything that’s not strictly a military target. Nuclear strike would probably change that.
I don't think there's any way NATO can afford not to respond to the offensive use of nuclear weapons in a war of conquest, just outside their borders, against a friendly nation. But as long as it's a small, tactical nuke, I think NATO's response will be conventional, but still big enough to cripple all Russian positions in Ukraine, and possibly more.
