Since WWI was called "The Great War" while it was happening, did those fighting and serving during the Second World War refer to it as "World War II" at that time?
Some may very well have, as the September 11, 1939 issue of Time magazine stated that “World War II began last week at 5:20 a. m. (Polish time) Friday, September 1, when a German bombing plane dropped a projectile on Puck, fishing village and air base in the armpit of the Hel Peninsula.”
Additionally, in 1941, President Franklin D. Roosevelt publicly alluded to the war as the “Second World War.” While most people would refer to the conflict as just “the war” or “the European war,” author Elizabeth Nix notes that “In 1942 he [Roosevelt] asked the public to propose alternate appellations, and over the next few weeks the War Department received 15,000 submissions ranging from ‘the War for Civilization’ to ‘the War Against Enslavement.’”
It seems pretty clear that the term “World War II” became more widespread as 1945 approached, as more and more references to a possible Third World War proliferated with the development and usage of the atomic bomb on the part of both the United States and the Soviet Union. Ultimately, by the time of Harry S. Truman’s presidency and the onset of the Cold War, the conflict was widely referenced as “World War II.”
It had a bunch of terms right off, of course, but there were at least some references to "the Second World War" or other "this is clearly the sequel" terms within the first weeks of the shooting starting.
The NY Times archive only has a handful of mentions in the 1936-1939 range, though most of those were in literary or academic contexts, as opposed to more popular ones.
Washington and Jefferson College actually established a Second World War studies program in mid-September 1939, specifically to try and figure out what was causing these stupid things and if they could be prevented. It referred to the conflict specifically as "the Second World War" even though it had been going on for less than two weeks at the time. Multiple faculty members were involved and teaching a series of courses on the propaganda and geopolitics involved. I don't have much other information on it at the moment, just the story of their announcement in the Times (13 September 1939 issue on page 3, for those with TimesMachine access).