One needs to be very wary about predictions of a coming "World War III," mostly because it has become a misnomer that often gets confused with a nuclear war or any holocaust scenario.
In reality, there was no such thing as "World War I" until 1941, when the bombing of Pearl Harbor by the Japanese got the United States to formally enter the war against the Axis Powers. With the USA added, the new situation got dubbed "World War II."
The previous international war was merely called "The Great War" until then. After the new "buzzword" for the 1941 war came into play, the Great War became "World War I."
Wars get their names after they've begun, and even then these can change depending on what follows or due to the final outcome. A nuclear exchange between the superpowers is often called "World War III" in the minds of many, but so are smaller nuclear war scenarios like the one we are talking about here. It has become an "anything to anybody" buzzword.
To be honest, even if North Korea started some nuclear exchange, it might not involve the entire world at war. In fact, some important academics and historians are checking to see if the title "World War IV" may be more appropriate due to words spoken by Pope Francis between 2014 and 2015.
Pope Francis has several times described the current world situation as a "piecemeal" world war, several times referring to it as "World War III" in his speeches to dignitaries while on his world tours. The Pope may be technically correct as almost the entire world is either engaged in the "War on Terror" or some other battle at present.
While only history will tell if the Pope will be considered the first person to dub the present world paradigm correctly as "World War III" (and I am NOT saying I agree with him whatsoever), semantically-speaking the current situation is more of a world war than an exchange of weapons between only the U.S. and North Korea would be. If anything, one might argue that such an exchange can only add to the current world paradigm, not be the act that begins or defines it.