The answer, somewhat boringly, is right at the beginning of (what we now call) the Second World War. That is, it was The Great War up until about the late 1930s, at which point thinking about these in terms of first and second sprung up almost immediately.

One of the only really useful things the field of Digital Humanities has produced so far have been tools that let you quickly map changes in terminology, notably Google Ngrams. So while the above conclusion is supported by hunting around in, say, newspapers from the time, you can also use Google Ngrams to quickly and immediately see such trends in a really unmistakeable way, by searching for the relative usage of these phrases across the Google Books corpus. Note that the differences between "the Great War" and "Great War" or "the Second World War" and "Second World War" were negligible (I checked).

The term "First World War" was first used in September 1914 by the German philosopher Ernst Haeckel, who claimed that "there is no doubt that the course and character of the feared 'European War' ... will become the first world war in the full sense of the word."

Shapiro & Epstein 2006, p. 329 citing a wire service report in The Indianapolis Star, 20 September 1914