Im not sure when the term 'World War II' was coined, but I'm assuming it was only coined that after the war ended or at least a few years in to it. What did people around the world refer to all the fighting that was going on?
It was called World War II right from the beginning, or the "Second World War" anyway.
That's because the First World War was already routinely being called "the World War" before WWII broke out.
Here's a post I wrote a few months ago that gives a rundown of when the First World War got that name, and how the Second World War got its name, too. WWII was being called "the Second World War" as soon as Germany invaded Poland in September 1939.