The term "world war" in the English language seems to have largely been an American thing.

As Elizabeth Nix writes, the British used the term "The Great War" for World War 1 well into the 1940s, with only few exceptions (Winston Churchill in The World Crisis among them).

The term "The World War" for World War 1 is from the American press, who had referred to the conflict as The European War, but when the United States themselves entered, looked for a new name.

As to World War 2, the British initially simply referred to it as "The War" or "The War against Germany". When in war, everyone of course knows which war you are talking about.

Nix writes that it was US President Franklin D. Roosevelt who named the war "The Second World War" upon America's entrance after Pearl Harbor. He himself didn't even much like the name and tried to replace it with something different (Nix does not state if that was for a particular reason), but his personal favorite, War for Survival, would lose out to the term "Second World War", which would for the sake of simplicity often be abbreviated to World War 2 after the war. During the war itself, "World War No. 2" was a Nazi propaganda term used in English-language German propaganda and was not really in use on the allied side.

With the impressions of the Second World War, the term of the former conflict was then extended to "First World War" or "World War 1".

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Sources:

• Churchill, Winston: The World Crisis (six volumes), 1923-1931

• Nix, Elizabeth: Were they always called World War I and World War II?, in Ask History, 2013

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Interesting tidbit: A German propaganda poster like the one I mentioned, hosted over at /r/PropagandaPosters.

To add to this question, what did they call it while it was happening?