The term grew more widespread as the scope and geographical extent of the fighting sank in and spread, but it was be being referred to as the "second World War" almost immediately in places. "The war in Europe" was a common term at the time as well, at least in the United States and prior to the Japanese entry.

The New York Times was using the term within a few days of the start of the fighting; the earliest reference I can find (off a casual search of their archives, mind, rather than poring over the full text of the papers!) dates to September 4, 1939 in a wire article from Britain. The Times' own writers were talking about "the first nine days of the Second World War" by the 10th.

At the same time, the nomenclature wasn't graven in stone yet. The Times' book review column for September 8 talks about "the outlook for books about what may or may not be known as the Second World War."

The Japanese officially called their portion of the conflict 'The Greater East Asia War', saying that it was a war of Asian countries fighting for their independence from the Western powers.

During the Occupation, the older term was banned from use, and 'The Pacific War' or, occasionally, 'The Fifteen Years' War' (from the beginning of the war in China to the end of the war in the Pacific) became the more common terms.