Russia using nukes only on Ukraine would get a whole new level of global condemnation and the loss of Chinese support, but would not cause the US to respond with nukes. That would be suicide for us.
We do not have an obligation to trade Kiev for Washington.
We do have an obligation to trade Moscow for Washington (and a thousand other places). The British and French have their own nuclear forces they could use separately, that is, not controlled by the United States. I think the British would rely on the US to decide, but France would absolutely launch if Paris was obliterated.
We would only launch ours if they launched a nuclear attack against us or NATO. That’s where mutually assured destruction (MAD) comes in.
I'm not saying the US will respond to Vlad's nuclear attack on Ukraine with a nuclear attack on Russia.
For one thing, we could do major damage without nukes.
I do expect that somebody would find a nuke somewhere, and Russians wouldn't necessarily know who or where it came from.
I'm saying the damage Russians would suffer from throwing nukes at Ukraine is incalculable, if not suicidal.
And for what?