I have answered this question before here, more or less, which gives more detail than I'll give here. The brief answer is: pretty much from September 1939 on. As you can see in my previous answer, there's a letter to the editor of the Detroit Free Press from May 1940 complaining about the popularization of the term "Second World War". There is another link in that previous answer showing an article from the January 1, 1941, edition of the Salt Lake City Tribune detailing all the events of "World War II" from the previous year--despite the fact that the U.S. was still more than eleven months away from joining the conflict. Another link shows a Gallup poll from April 1942 indicating that "Second World War"/"World War II" was far and away the preferred name for the conflict by the U.S. public at large, about four months into the U.S.'s involvement.

Similarly, British newspapers were using the phrase "Second World War" to describe the conflict by the end of September 1939, and World War II had been used a few times by the end of 1939 as well.

That all said, it was still quite common for newspapers to refer to the conflict as simply "the war" as it was ongoing, as many ongoing wars are. "The war in Europe" and "the war in the Pacific" would also be used, depending on the circumstances. But "World War II" and the "Second World War" were in common parlance by the end of 1939, certainly by mid-1940.

EDIT: I should also note that this question is addressed in the FAQ by /u/The_Alaskan. You can read that answer here.