All hell breaks loose. Almost.

Most likely there would be an overwhelming conventional response, beginning a no fly zone and escalating until Russia has retreated to it's own borders and Putin is either dead or extradited for war crimes.

That's assuming that it's only a small number of tactical nukes and no other countries or major population centers (eg, Kyiv) are struck. Nuclear wastelands are notoriously bad for business, so no one will want to make more. At the same time no one will want to tolerate the creation of any at all. So Putin would very, very quickly find himself with zero friends. Every nuclear weapon in the world (yes, even China) will be immediately aimed at Russia and fingers will be hovering over the launch buttons. No country will tolerate anything less then complete submission of Russia to the international community, probably represented by Nato (China might want a share though). Russia will be instantly and completely cut off in every possible way, by everyone. There will be no sanction busting, as everyone is 100% to preventing nuclear armageddon. This will continue until all demands are met, without fail or exception. Russia will collapse as a nation. It will be forced to surrender it's nuclear arsenal. It's territory will be divided up into regional puppet states of the other world power blocs. Russia will entirely cease to exist as a nation for the first time in hundreds of years. Putin will likely not leave to see it though, and if he does, it will only be in time for his trial and execution at the Hague.

If he launches at population centers or at any country other then Ukraine... things are worse. 30 minutes later, Russia turns to glass. And shortly thereafter so does a large portion of the rest of the world. Global warming suddenly becomes the least of our worries.

Putin undoubtedly knows all this. He might be a fool, but he's not literally insane. The odds of him using nuclear weapons are basically zero.