I have what seems to be the same problem. Photo Booth says no camera is connected, and FaceTime says it is receiving no video, and there is no green light, flashing or otherwise. However, this only happens when I log in as a user. Both FaceTime and PhotoBooth work when I log in as the administrator. (Giving my user account admin privileges didn't fix it.) On neither side do any programs show up in the privacy settings as having requested camera or microphone access.
I tried the "reset SMC" instructions and it didn't help. I think I may have inadvertently turned off the camera and mic for my user account when I upgraded to Catalina - if that is what happened, any ideas on how to fix that?