Actually, well before the war began. Not long after the end of the First World War, various publications that perceived problems with the Treaty of Versailles, or that just speculated on what a future war of similar scope would look like, referred to it as a Second World War.

I would also point out that this naming is by no means universal. Soviets called it the Great Patriotic War, and the Chinese term translates to World Anti-Fascist War.

I'll be honest, I'm not sure what the Germans or Japanese called their aggression at the time, though I suspect the Japanese acted like they weren't aggressors, and that there was simply unreasonable resistance to the Greater East Asia Co-Prosperity Sphere plan.