It's a very difficult to say yes or no because it's such a big event and there are many parameters to think of, but I'm going to do may best.

Let's begin with World War I. At the time of that war nobody knew that there would be a conflict afterwards, meaning that the term WWI wasn't used because of the fact that they didn't know that it would be the first of two major military conflicts. On the American side of the world the war was referred to as the "European War" by the American media. When the US entered the conflict in 1917 the term was replaced with the "World War" because the conflict was global in the matter of more than one continent was involved. The involvement of the Ottoman Empire also lead to the war being "global" and thus making the term "World War" even more accurate. Unlike the Americans the Brits used the term the "Great War" instead. That term was used all the way into the 1940's.

World War II is very similar to WWI in the sense of what it was referred to, namely that the conflict's name changed from country to country. In the Soviet Union for example, the war was called "The Great Patriotic War", often referring to the battles between Nazi Germany and the Soviet Union. The name "Second World War" was popularized by FDR when he put a label on the war in 1941. In Great Britain WWII was simply called "The War" until the late 1940's.

However, the term "World War II" was first used in 1919 when a British newspaper printed an article which predicted a future war, much like we predict World War III today.

The FAQs section has a number of answers related to this question.

According to Hew Strachan, in his work The First World War, Part One: To Arms!, whereas the British and the French largely dubbed WWI 'The Great War', it was the Germans that referred to it as Weltkrieg or 'World War'. This could be seen as a result of how the war played out: for Britain and France, the campaigns in Africa and the Middle East formed secondary considerations in Allied war aims, first and foremost of which was the defeat of Germany and it's Allies on the continent. For Germany on the other hand, the war quickly became one of Germany and it's Allies assailed by armies and resources from much of the world, while efforts to undermine the Allied war efforts embraced a 'global escalation' of the conflict: sending Lenin to Russia in 1917, supporting the Easter Rising in Ireland in 1916, encouraging Turkey to enter the war and the Sultan's declaration of Jihad against the Entente, supporting the Hindu and Muslim Conspiracies in India, sending agents to Afghanistan, Unrestricted Submarine Warfare, and acts of state-sponsored terrorism and sabotage in the United States.

There is no such thing as an official "World War." It is a name that many people (including heads of state) started to use to describe the conflicts of the mid-20th century. Similarly the "War on Terror" is a term that the Bush administration created to describe their actions (and indeed, many people disagree with the term and call it the "so-called 'War on Terror'" to indicate this).

As for the historical case of the "World Wars," the Google Ngrams database paints a plausible picture, in line with what other historians have suggested. In English you have people calling it the "Great War" or the "European War" in its own time. The term "World War" also starts to become common and by 1920 has eclipsed the term "Great War." The terms "World War I" and "World War II" both simultaneously become common around 1939, when Hitler invaded Poland and when the Europeans marked the beginning of another world war.