I was assuming a conventional war starting similarly to how it was in 1950. More on the nuclear scenario later. I think things will be different. If North Korea does in fact start it, I don't see China backing them up, at least not to the level they did in 1950. Nor would the Soviets be around to provide pilots and aircraft. China has a lot more at stake than they did in 1950. They can't be going around starting offensive wars, it would hurt their trade dramatically. Now if the U.S. attacks first, then I can see China backing North Korea up. Regardless, we would still have to to recover as we did in 1950 and retake lost land. If you look how long the buildup took in 1990, you will see why I say modern U.S. military is slower (compare to how fast the landing of Incheon happened in 1950). Of course we had nearby assets in Japan which we didn't have in Saudi Arabia, nevertheless it's not easy to move all the equipment modern divisions require. It isn't just men and a few tanks. It's a lot more than that.
Now since you mentioned Nuclear War, I can't see it happening. Despite fearmongering by the U.S. (who have their own agenda to push making Americans afraid), I don't believe North Korea is capable of delivering a nuclear device on any kind of missile. They can barely get a test one to explode. A bomb is easier to build, but of course they don't have aircraft that can get one anywhere. Any nuke detonation would result in Trump completely annihilating the country with nukes. And yes, I believe he would do it without hesitation. North Korea isn't that stupid. He would lose all the power he has by starting any kind of war, but especially a nuclear war. There is no chance America will be nuked by North Korea.