The Turks sending Syrian mercenaries to fight for Azerbaijan seriously raises the risks of escalation and the Russians being drawn in. It also ups the risks of Russia deciding to retaliation by supporting a Syrian offensive into Idlib.
I think it could get much worse.
What it really comes down to is how much Turkey is willing to support its cousins in Azerbaijan while risking their relations with Russia. Last time Russia and Turkey went head to head on policy, Turkey backed down. This may be different though. Although Armenia has lots of hills and mountains, making any sort of penetration into the region very difficult, it is essentially flanked on both sides. Nakhchivan would probably fall to Armenia on its own, but if Turkey is willing to go to the mat for this, the whole situation could morph very quickly.
Armenia can't possibly win a war against both Azerbaijan and Turkey, even if Turkey doesn't put in the bulk of its military power behind Azerbaijan. So Russia has to respond. Otherwise, you could very well not just see the contested region fall to Azerbaijan, but a corridor of territory falling to Azerbaijan that connects it to Turkey. Russia could probably protect the northern regions of Armenia with its forces in Georgia, but that would allow Turkey to fully extend its power into the Caspian Sea. Mind you, a sea that doesn't really go anywhere, but it would allow Turkey to possibly connect with other powers or even float a small navy.
The other issue is Iran. Because this is NOT something that Iran is going to want. If Iran decides to get involved, then that would cause Turkey to ally with Saudi Arabia, Israel, and other powers that are seeking to contain Iran. That in turn would probably force Iran and Russia to intensify their works in the other contested regions such as Syria, Lebanon, Iraq, and Yemen.
Turkey has already shown that it's willing to go to the mat to prevent a Kurdish nation from forming on its south. Are they willing to go to the mat for the sake of re-connecting to Azerbaijan? The Middle East has been coming to a hard boil for decades now. This conflict is probably going to push it closer towards spilling over.
In regards to NATO, that's a stickier situation than I think people are guessing. While I agree that neither America or the European powers are going to want to come to Turkey's aid if it starts trading blows with Russia because they want to invade the Armenians, people need to realize that this will probably spell the end for NATO itself. Sure, you can argue that Turkey is in the wrong. You can argue that most of NATO doesn't even really like Turkey.
But staying out of this fight will have the same result if the USA had not gotten involved in Vietnam; it will prove that the NATO members are not committed to the alliance. They will only show up if it's in their best interest. And that means that Eastern Europe will be left to its own devices if and when the Russians come for them. Ukraine wasn't technically a member. Turkey is a member, even if it's one no one actually cares for.
Now personally for me, that's not an issue. I think NATO has outlived its utility in the modern era. The US should in my mind, focus on keeping its presence in the Atlantic and the Pacific through alliances to the British and the Japanese (and friends) respectfully. We have no interest in this fight, so we should stay out of it.
