Thank you for being specific. Now, in response:
1. If NATO did violate its rules and inducted Ukraine while the war is still active, the Russian military would be crushed almost instantly. They have no way of competing with NATO air power; at absolute best they would inflict some painful losses before their air force and ground-based AA were wiped out. Doing so would take a few days to a few weeks, after which Russian artillery would be smashed to bits by air-strikes, and that's pretty much game.
2. I don't think NATO is going to have Ukraine join in the middle of the war, nor do I think they should. I think in the shitty situation going on right now, that massively supplying Ukraine in their fight against Russia is NATO's best move. It will probably result in either a stalemate or Ukrainian victory, and Russia ends up weakened and humiliated either way.
3. If NATO did join in the war directly, I still see the most direct result as being kicking Russia back across their border and stopping there, because Russia is a nuclear power and nobody in NATO wants to make them desperate.
4. In following of 3, I do not believe the Russians would actually escalate to launching nuclear weapons. Putin knows that if he does that, Russia will cease to exist. A humiliated Russia can recover and make another play at empire in 10-20 years, a Russia subjected to mass nuclear bombardment not only will cease to exist, it will go down in history as the most suicidal empire in human history, and its memory will be hated forevermore for all the destruction it wrecks.
You say you have no reason to think he wouldn't drop nukes, and I call utter bullshit. Mutually Assured Destruction has kept nuclear weapons from being employed in war since WWII ended, and nothing about its deterrent value against Russia (and China) has changed in that regard. Even if Putin lost his mind enough to try to push the button, you're presuming that his ruling cadre wouldn't stop him, and the fact that they don't want to die either gives them a pretty damned strong motivation for that.
If you want to make an argument that MAD no longer holds sway, you're going to have to do a hell of a lot better than 'I have no reason.' You need reasons, multiple strong ones, if you're going to convince people that Putin has become suicidal, and spiteful enough towards his own nation and people to destroy them as well.
5. Unless things do go nuclear, NATO joining this conflict would not expand it remotely close to enough to make it count as a 'World War.' Russia lacks the military strength to beat Ukraine by itself, Belarus is its only European ally, and their military is only a step or three above 'complete joke.' Given the competence displayed in Ukraine, the Russian military would be crushed very swiftly, and the conflict would be smaller than even Desert Storm in 1991.