Was the designation "World War" used during WWI and/or WWII, or was it applied afterwards. What would a soldier in either war have called it? Did people think of WWII as comparable to WWI while it was happening, thus warranting the "movie sequel" name (as a friend of mine put it)?
On an only vaguely related note, did soldiers tend to think of themselves as fighters in a "world war" or as fighters in a war against one or two other countries? Would an American soldier in the European theater have been aware of/connected to/informed about the Pacific theater? Would he have felt like part of the same war?
Google Ngrams is a useful tool for this kind of lexical tracing. If you plug in the phrases Great War, World War, World War I, World War II, First World War, Second World War you get a pretty straightforward graph that tells you more or less what you would expect to see: during WWI, they called it the Great War or the World War, and in from 1939 onward they started differentiating them as two World Wars that were connected to each other. By the end of 1945 the trends in referring to the wars had become completely stabilized.
As for fighters in the war — my impression is that American soldiers definitely considered the two theaters part of the same war. They did have news about the progress of the other theaters, which was distributed via newspapers and internal military publications.
hi! there may be additional info of interest in the FAQ*
When did we first start saying "First/Second World War"?
*see the link on the sidebar