While more can always be written, you may be interested in this previous answer of mine which talks about both the origins of the name of WWI and of WWII.

This sub's FAQ also has a shorter answer of mine which addresses the history of the name of the First World War alone.

TL;DR: "World War" had become a common name for WWI even before that war had ended, competing for dominance with "Great War". "World War" had won out in the U.S. by the early 1920s, a little later in Canada, and by the mid-1930s, it was probably the more popular term in the U.K. as well. In any case, even by the late 1910s, "the World War" was a well-understood term if not the only term used to refer to WWI in the English-speaking world.

In the years leading up to WWII, predictions of a "second World War" began to appear in the press. Immediately upon Germany invading Poland in September 1939, and France and Great Britain declaring war, the war was referred to as "the Second World War" and "World War II". Some people (particularly those who didn't want the new war to develop into a global conflict) did push for alternate names, but no other name ever caught on. From the beginning of that conflict, it was always called the Second World War/World War II.