/u/SpaceCamper3, this is a question that comes up from time to time here. You might try searching for other answers if mine doesn't suffice.

The first place to start answering your question is with another question: When did people start calling it the First World War?

/u/NorseWinter had a good answer to this question when he came up with Deutsche Kriegslieder, which was published in 1914.

On September 20, 1914, the Indianapolis Star ran a wire service report that includes the first known mention of the term "First World War." It quotes the German philosopher Ernst Haeckel, who claimed, "There is no doubt that the course and character of the feared 'European War' ... will become the first world war in the full sense of the word."

Now that we've established that the world wars can be numbered, this leads us to your question. If there was a first war, that implies a second (or third, etc.) The Oxford English Dictionary's etymology for World War 2 points to the Manchester Guardian of Feb. 18, 1919 as the first mention of the term. In that article, the Guardian was discussing the possibility of future wars arising from the upheaval caused by the Great War.

During the interwar years, there's plenty of speculative fiction that talks about the possibility of a Second World War. Heck, James Thurber's "The Last Flower" (published in 1939) mentions a "World War 12" as part of its antiwar satire.

What about the actual Second World War?

The first organization to label the war as we know it was Time Magazine, which used the term in its Sept. 11, 1939 issue.