I let Ubuntu upgrade to 16.04 today. Afterwards, the mouse would move, right-click worked, and left-click worked in some apps. Most notably it did not work on the system menu/notification bar. Doesn't even work on the login UI.
I knew this happened last time too, but couldn't remember why. I finally figured it out.
In Firefox, go into Settings, turn off chat. Reboot (I know, shouldn't have to do that, but I had to). Now, mouse works. Go back in and turn it back on, and it's still fine.