"Running late" is an idiomatic phrase that means, essentially, "late" or "being late," with the caveat that "being late" is more a translation than a phrase that people use. The Oxford English Dictionary defines it as:
- "s. to run late : to be behind schedule. Similarly to run early, on time, etc."
For your example, you could write "He was late so he sent a new doctor to run my follow up appointment." What's different about that way of saying it is that "running late" conveys information about why the doctor was late. He had a prior appointment and it took longer than expected. Perhaps it would be even better to write "He was taking longer than expected with a previous patient so he sent a new doctor to run my follow up appointment."
If you tell someone you are "running late" there are a few important connotations. First, it is probably in reference to a meeting or scheduled appointment. Second, you could already be late or simply anticipating that you will be late. That would depend on the time the message is conveyed relative to the appointment or scheduled time.
You are correct that "to run" the follow up appointment means approximately "to deal with." However, just know that at least in the United States, you don't normally "run" an appointment. It's understood, but not common. You might run a meeting but usually you "have" an appointment. I would have written it as "he sent a new doctor to meet with me for my follow up appointment.
I would also point you to a discussion on WordReference.com about this phrase.