I’m still not convinced that the doctor is a good person. Where’s he been all this time if the patient has been in this state for a month. The timing is too convenient.
The evidence is in his name 'Court Physician'. It was probably a pain in the ass to wrangle him free from the royal family he's most likely sworn to serve exclusively. Something innocuous as the common cold could've held him up depending on which family member it afflicted and if the king and/or queen wouldn't let him go until said family member was back to 100%. His prestige, pay, and probably his life are tied to his role so he can't just pencil them in until he's entirely free and with the go ahead from the royal family.