There is no simple way of explaining what happened to your mother. You can't call the doctors "first" and "second", especially if your mother wasn't in the room so she wouldn't know what you meant by that.
You would have to explain it in full by saying something like "When I got into the room, there were two doctors but only one doctor was free to see me. However, I told him/her that I wanted to wait to be examined by the other doctor."
This wouldn't be a likely scenario in the UK. A consulting room usually has only one doctor. Medical appointments are confidential so there shouldn't be another patient in the room when you're being examined.
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