I don't know how this hypothetical yet possible war would play out, too many variations to account, however I can say for sure something along these lines will happen:
NK will invariably lose. It can't fight the US alone, let alone fight a coalition consisting of at least: The US, SK, Japan, some European Powers, and possibly Russia and Australia. How it will lose and at what cost I can't predict.
If nuclear weapons are deployed, the entire world may likely have to deal with the consequences of a nuclear fallout. I don't know if NK missiles would reach the US before being hacked down, dissabled or shot mid-air, but it's possible that at least one detonation will occur "too close for confort". However, if nuclear weapons are used, we can be sure NK will no longer exist by the end of it, SK will be stupidly damaged or limping along, and the entirity of Asia and countries like Australia will have a to deal with nuclear fallout. Long term, the entire world will have an issue on it's hands due to radioactive contamination of the sea, and the spread of radioactive particles through both air, sea and animal life. However, it will not be the end of mankind.
If this happens to be a large scale war instead of a brief exchange of hostilities:
History will repeat itself and regardless if nukes are deployed or not, expect some kind of cultural "boom" that usually follows after a great power goes to war (see WWII, Vietnam, etc). Hostilities between countries might flare up, and the situation with Russia might escale, however since everyone and their mother has nukes today, expect a revival of the Cold War, kinda. Countries will resume the espionage and behind the courtains game trying to make whatever opponent it has less powerfull, and a lot of small or upcomming nations may be de-stabilized as a result, as it did during the Cold War when the situation was kinda the same.
Some third world countries may rise to become super powers, possibly South American countries because they'll be the ones least affected by war and political tensions, and currently they are still relatively stable and unaffected by what's going on, and may stand to profit as the current super-powers become dependant on them for subsistence. Some other less stable countries might break out into small scale war with it's neightboors, or into civil war due to popular unrest or some kind of revolution, and we may yet see another socialist-influenced dictator like Maduro because it seems the world isn't done making those. The Middle-East might become stable on it's own terms, not in a fashion that pleases everyone, however, globalism might start to decay as countries start to develop a strong sense of nationalism (as it always happens following large scale wa)
Eventually we will reach another period of relative global peace maybe up until the first decades after 2100's, if we are still stuck on this planet that is, until it comes a time where history sees fit to repeat itself and we end up on the same boat, or maybe we will fight Space Elves, who knows.
And if you thought fake news where bad, it will get worse as it will be irresistible to the media and governements to fear-monger the crap out of the fear of nuclear war, since that will invariably light up in the world again even if nukes are never used. Such fear-monger will be the basis and justification for other countries to meddle with one another, as well as the basis for economical or political sanctions.