This is certainly an interesting question but one that can never be answered accurately. Considering that there were more than 100 million individuals mobilized by the various international military regimes, you can see that there were at least that many different weapons used (assuming that all personnel carried a firearm). If you are talking about different makes of specific weapons, then the answer becomes more of an impossibility to gauge.
For example, German soldiers carried more than 40 types of handguns. There were at least 70 different rifle models used in the German military (excluding automatic and semi-automatic versions). Germans armed themselves with at least 10 types of machine guns and more than 40 varieties of sub-machine guns.
American firearms usage is a bit easier to determine. Colt and Smith & Wesson were nearly the only brands of handguns officially authorized by the U.S. military, while Browning machine guns were the only make authorized (in at least three different models).
Since many different nations produced multiple styles of weapons, the true numbers are practically endless. For a more detailed listing of weapons used by various nations in World War II, check the links found at the primary site listed below.