I think its going to be the continuation of Korean war, the second that the missile is determined to land in American territory, South Korea, or Japan, salvos will be launched by either three of those countries, which will lead to a quick peace treaty.
I think a country like China, if they would publicly ally themselves with N. Korea, wouldn't put troops on the ground but would supply N. Korea with weapons, etc. while facing heavy sanctions from the EU. Russia wont ally themselves with N. Korea, but they will sell guns to them without any NATO trouble.
I wouldn't say unit would be world war III in terms of battles, but either a proxy war (if the US isn't directly hit) or a continuation of the Korean war in which the US throws its full military support behind.