Similar questions get brought up in this sub pretty frequently, and I have written about it here, here, and here. The former two answers are largely based on the latter answer, with some updates and modifications. These previous answers address at least part of your question.

To summarize once again: contrary to popular belief, the terminology "Great War" wasn't very long-lasting, at least not in North America. The term "World War" was interchangeable with "Great War" in the U.S. and Canada while the war was still ongoing. "World War" first appeared in print to refer to the war on August 2, 1914, in the Pittsburgh Press, and became increasingly popular as the war went on. When the war ended, most American newspapers referred to the conflict as either generically "the war" or else the "World War" in their headline, while relatively few used the term "Great War" in their headline. A few papers split the difference and used the term "Great World War".

On the first anniversary of the end of the conflict, the U.S. War Department issued a press release, reported in the November 19, 1919, edition of the New York Times, which stated that the War Department would thereafter refer to the conflict as the "World War" in all official communications and documents. American newspapers followed suit, and by the end of 1920, "World War" was pretty much universal in its usage by the American press, while the term "Great War" had fallen out of favor. By the mid to late 1920s, "Great War" had fallen out of use in the U.S. almost completely.

In the interwar period, there were occasional predictions and jokes of a second world war, and so the terms "First World War" and "Second World War" popped up every now and again. The actual conflict known as WWII was first referred to as the "Second World War" as soon as Germany invaded Poland in September 1939, first in Time magazine, in their September 11, 1939, issue, and again a week later. Earlier that year, in June, Time had coined "World War I" and "World War II" in hypothetical terms, in predicting the outbreak of WWII.

By the end of September 1939, the term "Second World War" was ubiquitous enough in newspapers and on radio that a reader of the New York Herald Tribune wrote a letter to the editor, published on September 24, 1939, complaining about the usage of the term since the U.S. was not yet involved in it and it seemed like it might predispose the U.S. to being dragged into the conflict. A similar letter by a reader of the Detroit Free Press was published in May 1940. (See my second response above for quotes and details.)

By the end of 1940, a year before the U.S.'s entry into the war, the name was common enough that the Salt Lake Tribune ran a page-long article summarizing the events of "World War II" of that year. A Gallup Poll conducted in the first half of 1942 confirmed that the American public far and away preferred the "World War II/Second World War" terminology over any other term for the war.

In the U.K., the usage of the terms "World War" and "World War II" lagged behind their usage in the U.S. (Canada to some extent, too, but tracked closer to U.S. usage), but the terms were certainly not unknown. (Again, see my second response for more details.) By the early 1920s, the term "World War" was the term for WWI in the U.S., and certainly a term if not the more popular term in Canada. In the U.K., "Great War" was still preferred, but "World War" was a well-known alternative by the mid-1920s.

So "Great War" did have longer lasting popularity in the U.K. than it did in North America. While the term was pretty much completely out of fashion in the U.S. by 1930, in the U.K., it didn't fall out of fashion until the onset of WWII. In the aftermath of WWII, "Great War" was rapidly replaced in the U.K. in preference for "First World War" and "World War I", matching the American nomenclature that had essentially been in use for about 25 years at that point.

There was no such marked difference in the North American and European usage of "Second World War" and "World War II". World War II was referred to by those terms in the English language on both continents from September 1939 on, and wasn't really referred to in any other formal way by English speakers in any other way that approached those terms' popularity. Of course, the war was often referred to more informally and generically as "the war" until after it ended.

(cont'd...)