When did "World War 2" start being called "World War 2"?

When did World War 2 start being called a "world war" and when did it start being called World War 2? Thurber's The Last Flower (copyright 1939) makes reference to World War 12 so I'm curious as to whether he invented the sequential numbering of wars. According to Wikipedia, the book was published in November of 1939 (so one would assume it was written earlier) and the conflict in Europe was still relatively young and had not developed into a "world war" yet.

Another Wikipedia page says that WW1 "was predominantly called the World War or the Great War from its occurrence until the start of World War II in 1939, and the First World War or World War I thereafter." But there is no citation so I don't know if Thurber was following a convention within 2 months of the invasion of Poland or was creating a system of naming wars that others then followed.

OED says that the Manchester Guardian coined "World War No. 2" on 18 February 1919, "with reference to an imagined future war arising out of the social upheaval consequent upon the First World War (1914-18)."

Their next citation for "World War II" is Time Magazine on 11 September 1939.

The phrase "Second World War" appeared in Political Science Quarterly in September 1942; but it had been previously used by H G Wells in The Autocracy of Mr Parham in 1930.

It looks like Time magazine may have been the one that coined the phrase.

If this wikipedia article on the term is to be trusted, the September 1939 Time magazine article was the first reference to World War 2. WW2 was predicted well before it actually broke out and the magazine likely had the term prepared ahead of time. Time also has the first recorded reference to "World War 1", in June of 1939. There had been references to "The First World War" prior to that.