In Japan it was usually called the "Great East Asian War" (大東亜戦争, Daitôasensô), officially designated as such just a few days after the Pearl Harbor attack. It's important to remember, though, that Japan had been under a state of "total war" with China from 1937, which was referred to using the favorite euphemism of the day: the "China Incident" (支那事変, Shinajihen).

The interesting implication being, of course, that this war was "separate" from the one being waged in Europe.

You may be interested in this previous reply to a question about the name of WWI, in which the origins of WWII are also given. World War II was called "World War II" or "the Second World War" just as soon as Hitler invaded Poland in September 1939. By then, the first World War was already popularly called the "World War".

Maybe someone else can chime in about the war's name in other languages.