Only if you assume WW III would be US vs Russia
Relations between the US and Russia were getting a little worse, but they were nowhere near "early 60s" bad. Not even late 80s post-Glasnost bad. Helping stop ISIS doesn't prevent WW III either, because they simply aren't capable of fighting on that level.
Only four countries are truly capable of starting WW III: the US, Russia, China, and maybe Israel. If Trump makes relations with Russia better relations with China worse, and emboldens Israel to do whatever the hell they want without worrying about US backlash, I think it makes us more likely to get into WW III. But I don't think "more likely" means "very likely at all".
US and Chinese relations might suffer and go downhill, maybe with a few potshots over those stupid "islands" in the South China Sea they want to claim. Israel might start a regional war, which unfortunately will draw US troops into something that we should let Israel fight by themselves (on the logic that if you start a fight, you should finish it yourself, not expect your big brother to come over and clock the guy you sucker punched) Defend Israel if they are attacked without provocation, sure, but not if they start it.
But the chance of an all out WW III nuclear war? Very very small, even with a thin skinned snowflake in chief who can't keep his mouth shut. I firmly believe that if he ordered a nuclear strike that made no sense and/or was a huge escalation over the way a battle was currently being fought, the generals and sub captains who carry out the orders would refuse to do so.