It appears to have been used regularly by newspapers during the war.
I have a subscription to Newspaper Archive, which allows you to search hundreds of mostly smaller papers. On preview I see samclem beat me, but my search was deeper. 
Entering whole phrase World War II and limiting the years to 1942-1945 yields 60,360 hits.
The usage of “World War II” is exactly the way we use it today.
Chester Times - June 3, 1944, Chester, Pennsylvania
(Copied with all errors of scanning intact.)
The use started right away, with 6,653 hits from 1942 alone.
“World War 2” was less used, with only 879 hits from 1942 and 4,316 from the whole period 1942-1945. Interestingly, the phrase was often World War No. 2.
Modesto Bee And News-Herald, The - June 1, 1943, Modesto, California
“WWII” gets 1046 hits for 1942-1945. “WW2” gets only 44.
Going back before the U.S. got officially involved, the phrase “World War II” sees 2480 hits in 1941, 738 in 1940, and 105 in 1939.
It’s very possibly that many of the hits are duplicates of AP or UP syndicated articles so the number of usages would need to be determined with someone with more time on their hands than me but the ratios should hold steady. The OCR scanner and translator they use also sucks rocks. I tried for earlier uses than the 1938 Time magazine one. Each was a false hit, reading 'It" “He” “I” or even “Il Duce.” I did check enough of the later hits to ensure that the majority were true ones.
I see this as solid evidence that the European war was thought of as World War II as soon as it seemed imminent. That was before the Polish invasion. Articles talking about the possibility or inevitability of a World War II appear earlier in the year.
I don’t know why there would be such a split between radio and newspapers. Maybe a search of the relatively few dedicated news programs would show more.
No question, though, that people thought of the coming war as World War II from the beginning and it was already a common usage by the time the U.S. entered the conflict.