(or a member of an alliance that has 'a strike on any of us is a strike on all of us' security obligation) that response may not be a retaliatory nuclear strike. However, if the radioactive impact of a Putin strike damages a NATO partner retaliation would probably be open to reciprocal use of the biggest things on the table.
Retaliation for a nuke strike in Ukraine is harder to speculate on. If Russia tosses one of their many tactical nukes I think it's likely they toss many at easy soft targets. How many is a factor that would perhaps scale ( though likely exponentially) the international response.
Foreseeably responses would almost surely involve clean-up and health-care teams probably composed of militaryt and also probably protected by international military forces to deal with nuclear contamination and injuries. Also it's almost certain to result in the distribution of more and more powerful and long-range lethal aid, that could include weapons capable of reaching into Russian territory and vessels anywhere on the Black Sea.
I also would expect much enhanced economic and cyber warfare on soft but seriously important economic assets. It's likely that any nation standing with Russia would lose favored nation trading status. Beyond that are the many possibilities that depend on just how pissed off the international community becomes.
Crippling rather than merely weakening Russia would likely become the goal and Russian near full economic isolation would be advanced.