At what point was World War II known as World War II?

The best overview of the terminology that I'm aware of is a terrific essay on this in David Reynolds' anthology From World War to Cold War: The History of the 1940s, with the entirely appropriate title of "The Origins of 'The Second World War'." Much of this answer is sourced from it.

It is fairly widely known that the Soviet Union never adopted the term World War II, partially since the First World War didn't really fit into Communist ideology - for party theorists, it was a capitalist and imperialist struggle. Instead, the favored term used by the press and leadership starting the day after Barbarossa became "The Great Patriotic War," which tied in with the language used to describe the 1812 invasion by Napoleon ("The Patriotic War".) I can say that long past the fall of the Berlin Wall - and back when I still comprehended Russian on more than a very marginal level - that terminology (Velikaya Otechestvennaya Voyna) survived well into the 1990s even by Russians far outside the FSU, and I believe it's still used today.

For the Chinese, the preferred term is apparently Kang-Ri Zhan-zheng, or the War of Resistance Against Japan, but I don't have first hand knowledge of the scholarship here, although it looks like it may have been a post-1949 event. I'm not sure if the current regime has refined this further.

The Japanese had a slightly different take; I'll quote Reynolds directly here because it succinctly summarizes the complexity of the multiple political implications underlying the language so well.

"Its brutal and massive invasion of China remained an undeclared war and was therefore dubbed, in a characteristic Japanese euphemism, as ‘The China Incident’. After the conflict expanded in December 1941 to include the United States and the European colonial powers, it was described as the ‘Greater East Asian War’ (Dai Toa senso). Under the American occupation after 1945 the ‘Pacific War’ (Taiheyo senso) became the official title, but nationalist revisionist writers revived the earlier term in the 1960s."

For the French, de Gaulle referred to the war as "une guerre mondiale" ("a world war") after 1940, but his audience was small and I don't have any references for what the Vichy called it.

For the Germans, the most frequently used general term was der Krieg ("the war") although this sometimes was escalated to Weltkrieg ("world war"), which fit into the vision of it as a continuation of the war that had begun in 1914 since the latter term had been part of the contemporary vernacular for the previous one. (This also dovetailed with the Nazi straw man ideology of worldwide Jewry as the underlying enemy.) Postwar, it was relatively easy then to adopt the English language terminology - Zweiter Weltkrieg.

Now we get to the English language fun.

In Great Britain, there were a couple of early references. One was from Duff Cooper, one of the tiny handful of Tories in leadership who openly opposed appeasement, outright resigned after Munich, and later served in various posts under Churchill. Much like Churchill, Cooper had made a majority of his income by writing, and in late 1939 after war was declared he tried to make a quick buck by publishing a collection of his previously published criticisms over the last couple years as a book titled "The Second World War: First Phase". It didn't stick; like their German enemies, pretty much everyone else on the British side of things just called it "the War."

Churchill made a comment in August 1940 talking about "this second war against German aggression" and finally talked about "a great world war" in July 1941 and after Pearl Harbor referred to it in front of a Joint Session of Congress as "Twice in a single generation the catastrophe of world war has fallen upon us," but there even was a question asked by a publisher of His Majesty's Government all the way in late 1944 about what to call it given a discrepancy between the American and British references - and no official response was forthcoming, so "the War" continued in popular vernacular.

That finally changed a bit when Churchill wrote his memoirs. His initial working title in 1946 was "The Second Great War", but finally in September 1947 when the initial release of his books in the United States market was just a few months away, he formally settled on "The Second World War." As I've written before, given the unexpected prominence of Churchill's memoirs in the immediate post-war literature, I'd argue this had an outsize role in the wider English language adoption of that terminology. With official government war histories well underway by not just the British but also the Dominions and others, all involved were polled in 1948 and - with the exception of Australia, which apparently disagreed - the concession by Churchill to American standards was formally adopted across the English speaking world under the leadership of the Attlee government. French government entities formally followed suit by late 1950.

And last, we have the United States.

As the previous answers of /u/lord_mayor_of_reddit and /u/the_alaskan point out, you can trace the first widespread popular usage of "World War II/Second World War" back to September 1939 and Time Magazine.

But there's some massively ironic context here in that Time of all publications - publisher Henry Luce and his wife and future Congresswoman Claire Booth Luce had been among the most virulent enemies of Roosevelt administration policies both foreign and domestic from 1932 onwards - unwittingly helped FDR in his attempt to change the minds of the American people about the European war and how it would affect their lives. FDR's struggle against isolationism from 1939 through 1941 is far too broad a question for this answer, but he began to refer to the conflict as 'a world wide war' in May 1940. On March 8, 1941 - notably, the day after Lend-Lease passed - he expanded it further to state outright, "When the second World War began a year and a half ago" and continued this usage throughout 1941. This appears to have been a deliberate choice of language to get the citizenry of a reluctant United States thinking about an overseas war that would affect them even if many clung to a stubborn belief that it wouldn't or shouldn't.

When it came time for an official government title of the war, though, FDR balked at his own choice and in the Spring of 1942 stated publicly at a press conference that "the Second World War (wasn't a) particularly effective (title)" and even solicited public suggestions for the War Department to come up with a new one. (This was the reason behind the Louisville Courier-Journal article that /u/lord_mayor_of_reddit posted on the subject.) He briefly tried "The Survival War" (it didn't translate well and some aides thought it sounded desperate) and "Everyman's War" and even as late as 1944 thought about a term he'd heard in passing, "The Tyrant's War." None of these attempts even made a ripple, so FDR's unintentional intentional choice of language is how the Western World found itself with the interchangeable terms the Second World War and World War II.