I don't think so. I think a really important thing to bear in mind is that these speeches are also designed for domestic consumption. It's really essential for Putin to persuade the Russian public that the West poses an existential threat to their country, and talking about the threat of nuclear war is a good way to keep people believing the war is necessary. Ordinary people in Russia will be just as terrified by the idea of nuclear war as we are.
In fact, I remember reading an interesting study carried out after the Cold War that studied the fears of teenagers in Russia and (I think) the US. The most common fear in American teenagers was the death of their parents, with nuclear war second. The results were the opposite in Russia- teenagers there were most afraid of nuclear war. Those teenagers are now adults who will remember that terror, and I suspect playing on those fears is partly a control tactic. I don't know if you've ever watched Russian state TV (I don't recommend it if you're nervous) but it's full of outlandish discussions of Russia's nuclear capabilities and contemplation of a nuclear conflict. This is for domestic viewing, so it must serve some purpose, in Putin's mind at least, to keep ordinary Russians afraid of nuclear war.
I also find it comforting to remember that we've been very, very close to nuclear war before and always stepped back from the brink. Nixon actually sent nuclear-armed bombers to fly directly at Soviet airspace during the Cold War to try and intimidate them and we're all still here.
Putin was always going to rattle the sabre- I'm actually surprised that he hasn't rattled it more. But as far as I'm aware, there has been absolutely no change in Russia's nuclear stance beyond some fighting talk from Putin. I'm not saying we shouldn't take the threat seriously, but we've been here before, and here we are still alive.
If you're anxious, I think it's best to disengage with it if you can. He's rattled the sabre before, and he's going to do plenty more rattling IMO. Best to be prepared for that and try not to let it get to you too much. It is unnerving but it isn't unexpected. He's trying to mess with your head- don't let him!