Time magazine called it World War II in the september 1939 issues. London newspapers used the same headline (second world war) on the very day the uk declared war on germany.

The thing you need to remember is people had been talking about a hypothetical 'second world war' since about 1919 (the manchester guardian asked if the russian civil war would become it and you had fiction writers write about it).

So at least some of the press jumped the gun to that term as soon as something that looked likely happened. I think as a widespread term that only came later.