Mutually assured destruction is still a thing. We may not be at Cold War levels of insanity, where between the US and Russia there were enough nukes to glass the planet like 150 times over, but plenty of nations have arsenals (especially in Europe), and the best way to make enemies of the entirety of the world would be to be the first one to launch a nuke. Dropping a nuke would signal to every leader in the world that no country is safe from becoming an irradiated wasteland.
I think if Putin dropped a nuke, his allies would drop him faster than it would take NATO to declare all out war with Russia.
Unfortunately, nobody would be able to take him out that quickly. Russia still has plenty of nukes, and they could fire them all before anybody has time to react. If that nuke is an ICBM, though, as soon as it leaves the silo the world would know, and the counter barrage of nukes would be firing up before it even lands.
I originally meant that dropping a nuke would have the entire world declare war on Russia, even his former allies because no one wants to rule over a pile of radioactive rocks, but thinking about it, his allies would probably be the ones most likely to try to have him assassinated in that situation. A maniac with a big stick is only useful so long as you don’t have to worry about him smacking you with it, too.