When was World War 2 named, officially? I know that World War 1 was the “Great War,” and naturally when what we now call WW 2 started no one was expecting another Great War.
It was also called the WWI was also called the World War.
I listen to a lot of Old Time Radio (OTR) and I don’t ever recall them saying World War II. They say the first World War a lot, but off hand every thing I recall the people on the radio say “the war in Europe” or “the war in the Pacific.”
Considering Poland wasn’t invaded till 1939, the term WWII was probably used for a future thing, kind of like “If Iran gets 'the bomb” it’ll be WWIII."
Mildly related:
A Swiss-French friend told me that WWI was called “le guerre de 1914 à 1918,” the war from 1914 to 1918, but WWII was “le Deuxième Guerre mondiale,” the Second World War. Funny that.
What do you mean by “officially”? Language is arbitrary, so people name wars whatever they want, and eventually one name starts to be used more than the others, at least in a certain region or language. If you mean to ask when the government officially decided upon the name of the war, perhaps for the purposes of veterans’ benefits, you’ll have to specify which government, or maybe even which branch thereof, you’re talking about. Not all countries refer to World War II as such; the USSR and its successor states, for example, refer to it as the Great Patriotic War.
How about “When did World War II become the unmarked phrase in English to describe the military conflict of 1939-1945?” We’ve established that the phrase was used long before hostilities broke out, and I agree with Markxxx that 1940’s mass media tended not to use it. Perhaps it picked up steam in America during the next overseas conflict?
It was used extensively by the media(judging by a search of my newspaper database) in the US starting around 1940-41.
The 1914-1918 war was “World War I” because elements of it were fought in China and Micronesia (Japan conquered German colonial holdings) and Africa (Germany’s four colonies were invaded by the allies; Luttow-Vorbeck held out in central German East Africa (now mainland Tanzania) until after the November 11, 1918 armistice). In addition, the extensive holdings of the Ottoman Empire in Mesopotamia, Palestine, Syria, etc., were overrun by Allied, mostly English, forces.
It might be worth noting in this regard that in Churchill’s History of the English-Speaking Peoples, he entitled the chapter dealing with the Seven Years War (1756-63; “French and Indian War” in America, and the war that consolidated English rule in India) as “The First World War.”
Newspaper Archive results for hits on “World War II.” N.B.–many hits will be duplicate AP stories, etc.
1940 775 hits
1941 2500 hits
1942 6200 hits
1943 8000 hits
1944 13,600 hits
1945 31,600 hits
1946 58,000 hits
1947 46,000 hits
1948 46,000 hits
1951 82,000 hits
1956 55,000 hits
That’s interesting, but being a teenager when the US entered the war, I never recall anybody using the term “WWII” until well after it was over. The media may have used it, but it certainly was not commonly referred to in conservation. At least, where I lived at the time.
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.