Would quickly accepting Russia and Ukraine as NATO members end the war?

As rossdavidh mentioned, a country may not join NATO while at war even if one calls it a military operation. If either or both countries joined while at war then all NATO countries would be obligated to join in, thus solidifying a full scale WW3.

So I think your question is really how long would it take for WW3 to end and I guess that depends on tactics used. Probably less than a few years and I have no idea whom or what would remain.

1) one requirement for NATO admission, as I understand it, is that you don't have any active wars going on.

2) while there was one case where Turkey and Greece went to war when they were both NATO members, generally speaking it is a requirement that you don't attack other NATO members.

3) it is expected that all NATO members must be democracies, and while Russia has had elections, and I think arguably was a real democracy in Yeltsin's day, it hasn't had competitive elections in decades. You don't have to be a perfect democracy to be in NATO (otherwise Turkey and arguably the US might not qualify), but you do have to approximately speaking have elections in which the incumbent might possibly lose (true in both Turkey and US today, not true in Russia).

4) What Putin actually wanted was to reassert Russia's dominance over Ukraine, making it somewhat similar to Belorussia in that it would have a separate government, but would not be really independent. The stuff about worries about Russian security was a pretext, kind of like WMD's were the pretext for the US in the Iraq War but not the real primary reason.

If Russia returned the annexed lands to Ukraine the war would be over. No need to join NATO (which is a hilarious/insane idea). Just war reparations to deal with. They will be large. Very large. And we will see to it that they are paid in full, with or without Putin as the dictator de jour.

Russia would have to want to join, which it doesn't because (at the very least) NATO is it's existential 'Other' threat that justifies it's politics.

NATO was created to fight against Soviet Union/Russia so I don’t think that this will be a case. Long time ago Putin they’ve asked him about it in the interview and replied that he asked Clinton (I think) and he just laughed.

No one knew his plans for a fact but since NATO betrayed him multiple times and growing up in the SU he probably romanticized it a lot which eventually created this situation.

If everyone is in NATO who are they gonna be fighting against then?