From my tests on my vWii, the original Wii U Pro Controller worked normally in homebrew applications, just like it did on the Wii. It worked on Nintendont and emulators without any issues. On the other hand, when I tested a generic Pro Controller, it didn’t work.
I synced the original Wii U Pro Controller the same way we sync a Wii Remote. On the Wii Menu, I opened the Home Menu and pressed the sync button on both the Wii and the Wii U Pro Controller. The Player 2 battery icon in the Home Menu blinked white and then turned off again, indicating that the controller was paired but is not compatible with the Wii Menu (which is why it didn’t stay connected).
When launching any homebrew application, whether an emulator, Nintendont, or any other, pressing any button on the Wii U Pro Controller made it connect to the Wii normally, and I was able to use it wirelessly without any issues. On the Wii U, it was even simpler since the Pro Controller was already paired with the Wii U, so I just had to open an application for it to connect.
As for generic controllers, none of the ones I tested worked. Incompatibility, perhaps?