Unless Russia counters Turkey's move by invading Turkey itself, WWIII's not happening, because NATO primarily commits to defending the territorial integrity of its members against foreign aggression. I don't think anyone believes that Syria is part of NATO, and Russian defensive actions against Turkey in Syria aren't exactly foreign aggression.
What might happen is that the Russians might test NATO's resolve by flattening Turkish air bases with ballistic or cruise missiles. My suspicion is that NATO won't respond with missiles against Russian air bases.
That's a key point right there. If Turkey invades Syria and Russia then responds, NATO can argue that Turkey started things, and so the mutual self-defense clause does not apply.
Just the opposite - You are forgetting the Russians have their most advanced weapons in place for which there is no counter as yet. Resupply through Northern ports and by air over Iran - where most supplies are coming in from now.
There are no US/NATO missiles which can currently get past the S400 and S300 bateries. No planes can get close enought to the coast to launch stand off missiles.
The Turks will lose control of the Kurds and disaster will follow.
And if that doesn't do it, don't be surprised if Russia arms these missiles with tactical nukes - Putin has already hinted that it might become necessary to use them. And, IMO, Putin will do what ever it takes to keep Syria from becoming an Islamic state - even if it means the total destruction of Syria (Isaiah 17).