I can see a situation whereby the Ukranian armed forces fight Russia to a standstill (which appears to more or less already be the case), and logistical issues and financial pressures become so intolerable that Vlad, realistically, can't continue to prosecute the war conventionally to any worthwhile effect. In that case, I could see Russia making a show of using a singular tactical nuclear device, hopefully in an unpopulated area, in an effort to send an 'enough is enough' message to Ukranian resisters and draw the fighting to a close without the shame of having to admit a strategic defeat and wind down the conventional effort.
None of that need precipitate a full exchange between Russia and NATO though, because even if it does come to pass nobody in the West is going to be sufficiently inclined to use their own nuclear capability on Ukraine's behalf.
So no, while I think nuclear deployment is certainly possible, I don't think it's in any way likely to escalate into a full WWIII scenario. What I think the past month has shown though is that this truly is the culmination of 50-60 years of steady decline in Russia's credibility as a global superpower. If Vlad truly does have designs on taking on NATO as some suggest, and I don't believe he does, then that conflict is evidently going to be fought inside Russia itself, because right now I wouldn't back his armed forces to make it over the border if they tried to invade a NATO/EU member. The Baltic states have always been vulnerable to pretty much what is happening to Ukraine, and if he was going to try it on with NATO anywhere I'm pretty certain it would be there, but I think the past month will have been a rather sobering reality check for Putin which will have tempered any designs on escalating this. Right now, Vlad's only real issue is how to extricate Russia from this shambles he's created without losing so much stature that he's deposed from within, or triggers another 1990's-esque internal struggle for power within Russia itself.