You’re the receptionist. What would you have done?

Surely it would have been better for you to have gone first to save time. The late person might have been a no show. Not fair for you to be penalised for not being late . Receptionist should have shown a bit of common sense!

Impossible to say without knowing why the first patient was late. May well have been circs entirely beyond her control and that she had phoned ahead,

Were you waiting for 15 minutes whilst the consultant had no patient in the room, or did somebody else come out after 15 minutes, then the patient who was late went in?

Normally a late patient, unless held up by an ambulance taking her, would wait until there was a gap such as a no show. You should not have been penalised.

I would be amazed if she did ring through because normally it takes ages if at all.

Zurely, the receptionist would be acting on the instructions from the consultant or the accepted protocol

If the room was empty until the latecomer arrived, then you should have been allowed to go in first.

It’s difficult.
Did they know she was going to be 15 minutes late?
IMO You should have been seen first.

If they didn’t know how late she’d be and the 5mins stretched to 10, then 15, Thats harder to decide.
I hope your visit had a satisfactory outcome when you were eventually seen.