In the uk most doctors do not have such a system or list, if you miss an appointment and do not reschedule within a reasonable time beforehand, you have to reschedule yourself after the appointment. From time to time some doctors or their staff might make or offer you a new appointment, for elderly patients, pregnant women etc or patients with a serious condition. Some surgeries put aside a part of a day or a time slot after surgery for walk in appointments. A walk in appointment is where you turn up at the surgery and wait in a queue [British] line [American] and you will be seen by a duty doctor or any doctor not necessarily your own doctor. If you arrive late for an appointment then another person will be seen and the the next one after that and if the doctor gets through the appointments in a reasonable time you might be called in later, or you might be called in at the end of the surgery time, other patients who arrive on time will not be pushed back because you were late. For elderly or disabled patients or patients that have to wait for A NHS bus or Taxi that is late, the doctor will know about any delays and the surgery or hospital will usually accommodate you when they can on the same day.

It's definitely not 'pass numbers list'. I don't think there's a specific term for the scenario you describe - at least, not where I am. Numbers just aren't handed out for a missed appointment. In my country if you arrive late and miss an appointment by 10-15 minutes, then the receptionist usually will say, "I'll try and fit you in next". If I arrive later than that, then they reschedule me for the next time available and give me the option of going on a 'waiting or cancellation' list. If someone else cancels, I get their spot. If I've completely forgot the appointment that day and I phone to reschedule, they'll usually book me in for the next available appointment AND they'll ask me if I want to go on a cancellation/waiting list. That is, if someone cancels an earlier appointment, I can take theirs.

if I were your teacher, and I make things up to teach my students, I would call it the "goodness" list as in depending-on-the-goodness-of-the-receptionist list.

Thanks

A general term is a "wait list" or a "waiting list." When students register for classes, some students are put on a "wait list," if there are more students than seats. These students are waiting for open seats (which occurs when some students in the course change their minds and drop the course).