The Wheelwriters used membrane buckling spring keyboards (they were in fact the first IBM product to do so)
Differences in sound can largely be explained by the construction of the keyboard, both in terms of the fact that it is mounted inside a typewriter instead of plastic shell, and also that it's rather old (they all sounded slightly different over the years) I also seem to recall that the Wheelwriter keyboards have some sort of dampers on them which may affect their noise vs. a regular Model M.
To answer your question more directly - the most straight forward way to check is to remove some keycaps and look at the top of the keyboard assembly (sometimes called the barrel plate). If it's all plastic, it's an M. If it's metal with plastic barrels, it's an F. If the backplate is attached to the barrel plate with plastic rivets, it's an M. If they slot together at the top and bottom with metal tabs, it's an F. The presence of a ribbon cable also suggests an M, I think every Model F assembly I've seen had the controller permanently attached to the assembly with a non-removable flat cable.
EDIT: There were a few obscure Model M variants which used a Model F style assembly, so the presence of a plastic ribbon cable instead of a flat cable is probably the most reliable indicator.
There's no mention of how to service the keyboard in the manual. I could try and dismantle it, but I could damage some of the ribbon cables in the process, which I do not want to do.
With both the Model M and Model F, you were not intended to dismantle the keyboard assembly, instead you'd toss it and buy a new one if there was something wrong with it. In the case of the M, it is simply not designed to be opened. For the F, IBM seemed to believe that there was too much risk of contamination interfering with the capacitive sensor card.