This has been asked and answered before, more or less. Contrary to popular belief, the term "World War" was interchangeable with the term "Great War" while the conflict was still ongoing, particularly with the North American media. WWI was first called the "World War" by the newspaper the Pittsburgh Press on August 2, 1914.

Perusing American newspaper articles during the period of 1914 to 1918, it's not difficult to find mentions of the conflict as the "World War", as typified by publications of columns from the St. Louis Globe-Democrat and New York World. Same deal in Canada.

The first time WWI was called the "First World War" was probably by Laurence Stallings' 1933 photography book "The First World War" which was kinda/sorta turned into a movie of the same name the following year. The film, in fact, was comprised of silent newsreel footage of the war narrated by author Stallings. The term "World War" thereby continued to gain popularity during the interwar period.

So, when Germany invaded Poland in the fall of 1939, it wasn't out of character for the American press to begin using the term "World War" right away. The first instance of the use of "Second World War" comes from the September 11, 1939, issue of Time magazine. (A post-WWI conflict had been referred to as "World War No. 2" as early as a February 18, 1919, article in the Manchester Guardian, but that was still hypothetical, and not a reference to any actual war that had yet been fought).

Thus, the terms "Second World War" and then "World War II" were used from September 1939 right until the end of the conflict.

Here is a letter to the editor of the Detroit Free Press from May 14, 1940, in which the writer complained about use of the term "second World War" since the war was still firmly being fought in Europe. The writer was worried that the terminology would get the U.S. involved in this "world" conflict.

Here is an article from the Salt Lake City Tribune on January 1, 1941, recapping the events of "World War II" from the previous year, which the U.S. was still not yet involved in.

And here is an article penned by George Gallup (of the Gallup Poll) that ran in several papers across the U.S., here printed in the Louisville Courier-Journal on April 29, 1942. According to Gallup's polling, "Second World War" and "World War II" were the preferred terms for the war by the American public, though FDR and other government officials preferred different terminology.

So to answer your questions:

"At what point did World War 2 become known as World War 2?"

September 1939. It started in the American press and gained popularity from there.

"When did the Great War become World War 1?"

It became the "World War" right from the summer of 1914. First World War came into common usage once "Second World War" was coined in the fall of 1939, though it had been used occasionally during the interwar period. By the end of 1939, the terms "First World War", "Second World War", "World War I", and "World War II" had all been used in print.