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The general term 'World War' has been in use since at least the late-19th century, so this would not have been a new concept at the time of The Great War's outbreak in 1914. However initially the First World War was viewed as a European conflict, known in the US at once point as the European War. It was only later in the war that it became a more global conflict and the US began referring to it as the World War.
Supposedly the usage of the phrase 'World War One' began even before the end of the war itself with Colonel Charles à Court Repington credited with coining the phrase in September 1918 in his diary at the time:-
"To call it The German War was too much flattery for the Boche. I suggested The World War as a shade better title, and finally we mutually agreed to call it The First World War in order to prevent the millennium folk from forgetting that the history of the world was the history of war."
Repington went on to publish a book in 1920 called 'The First World War' (from which the above is a quote).
Fox later produced a film in 1934 entitled The First World War, in turn based on a 1933 book by Laurence Stallings, 'The First World War: A Photographic History'.
Nevertheless, reference to the First World War would not have been in common parlance during the inter-war period, and generally it continued to be referred to as the Great War, or simply the War.
The phrase 'World War I' (as opposed to First World War) first appeared in the June 12, 1939 issue of Time magazine when the same article referenced the possibility of the imminent outbreak of World War II. Following the outbreak of the Second World War later that year, Time magazine again referred to the Second World War in its September 11 issue.
However, this was not the first reference to a Second World War. On February 19 1919 the Manchester Guardian published an article about the idea of a future 'Second World War', and this is what the Oxford English Dictionary credits as the first usage of the phrase the Second World War.
Despite this, during the Second World War itself generally the conflict was referred to as, again, simply 'the War', particularly in England (as opposed to the US) and was not commonly called the Second World War at the time.
One interesting contemporary example of this was in April 1942, when the then US president Franklin D. Roosevelt discussed what the current war should be known as:
"So I am looking for a word-as I said to the newspaper men a little while ago-I want a name for the war. I haven't had any very good suggestions. Most of them are too long. My own thought is that perhaps there is one word that we could use for this war, the word "survival." The Survival War."
More than 15,000 suggestions were made by the public to the War Department following this, but none stuck.