How likely is Nuclear War?

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What do you mean by that? Let's not get too hung up on the old concept of MAD - just because one nuclear missile is used does not automatically mean every nation-state will shout FIRE ZE MISSILES and end human civilisation in a catastrophic nuclear exchange.

I actually think there is a relatively - and worryingly - high chance that the current brinkmanship will lead to a nuclear missile being used in an aggressive manner, even if is is the DPRK launching a missile into US territorial waters to make a power move. There is a smaller, but not negligible chance of nuclear missiles being used on the peninsula itself; and a very small (tiny even) chance that PRC and USA will be sufficiently at odds that a nuclear war develops between them.