When I was a clinic assistant and in my current role as a student, I have done my best to kind of “run interference” by getting some portion of the next appointments done to give the physician more cover and keep the next patients from getting too mad about the wait. I also give an explanation with my apologies, saying something like “we had a bit of an emergency come up”, or “the previous patient ended up needing more time than we had scheduled” while apologizing for the delays.