There are a variety of specialists who hold medical appointments to the Queen or the Royal Household - Physician to the Queen, Surgeon to the Queen, Orthopaedic Surgeon to the Queen, Physician to the Royal Houshold, etc. They are all leading consultants in their field, with busy private and public practices. The appointments are largely honorary, with actual service provision arising only occasionally.
The current Physician to the Queen is Professor Huw Thomas FRCP, whose day job is as a consultant physician at St Mary’s Hospital London, and Director of the Family Cancer Clinic at St. Mark’s Hospital. Apart from providing consultant physician services as required, his role also involves advising on who should be appointed to other positions in the medical household.
There are two Apothecaries to the Household, one in London and one in Windsor, and they run daily or near-daily GP clinics at Buckingham Palace and at Windsor. So there are GPs in attendance at both places for at least a couple of hours on most days. Note, however, that the Queen is often not in attendance at either place - she may be at Sandringham or Balmoral, for example. There is, I think, a full-time nurse on the staff at Buckingham Palace.
When the Queen travels abroad, her entourage normally includes a medical officer, traditionally a surgeon from the medical service of the Royal Navy. So she is well-positioned to have her wounds treated, and to receive advice on the transmission of venereal diseases.