So, just to make the quick correction — "thermonuclear" means nuclear fusion temperatures. "Nuclear" is ambiguous — it can mean fission or fusion. The nuclear weapons used in World War II were 100% fission weapons, not fusion weapons (the "hydrogen bomb" is a thermonuclear weapon, and uses fission reactions to induce fusion reactions which in turn induce fission reactions — the first H-bombs were on the order of 1000X more explosive than the weapons used in World War II). So you'd never call WWII a "thermonuclear war," but you might call it a "nuclear war."

So, that aside: was World War II a "nuclear war"? It depends what you mean by "nuclear war." If you mean, "a war in which nuclear weapons were used" — then yes, it was a nuclear war from August 6th onward. If you mean, "a war in which more than one party uses nuclear weapons," then the answer is no, because only one party was nuclear armed.

Usually "nuclear war," to most people and scholars, means the latter. "Nuclear war" with the Soviet Union would have meant a bi-lateral use of nuclear weapons. This is the "nuclear war" that people feared in the Cold War, not the use of nuclear weapons against non-nuclear states.

Is this inclusive of the definition? No. Some have argued that we should talk about World War II as a nuclear war, too, because of Hiroshima and Nagasaki. So my friend, the scholar Michael Gordin, has a book called, Five Days in August: How World War II Became a Nuclear War (Princeton University Press, 2007). His argument is, however, about exactly this question: at what point did people consider nuclear weapons to be in a totally different category from "conventional" warfare? In fact, he argues, that very question was being deliberated in between the time that the bombs were used (which was seen by many as just an escalated form of conventional warfare) and the surrender of Japan (which cemented the nuclear weapons in the public and official eyes as "special" weapons different than anything that came before — a weapon that could not only end one war, but might end all wars). (I reviewed his book if you want to know my take on the argument.

So that's the clever way to make the argument, to ask why we use the term "nuclear war" as a special category in the first place, and exactly when that "difference," that "specialness" got created. If you call World War II a "nuclear war," you are drawing special attention to the technology of the atomic bomb — which you might want to do (maybe you want to draw attention to the role of the bomb in the war, for example), or might not (the vast majority of World War II's fatalities, even the fatalities from bombing, did not stem from nuclear technology). When we categorize things, we are making an argument. The trick is to make sure you are doing it with an awareness that you are doing it, and to be able to defend the choices you have made.