Wow! what an amazingly narrow and blinkered westernised view of the world, that is not how it works everywhere!
100 years ago in the uk surgeons were apprenticed, no reading or writing necessary, and in many places around the world it is still the same.
You are an illiterate 10 year old with no education. You help carry your critically injured father to hospital after he stands on a land mine. You are the only able bodied person in the room when the surgeon says he needs another pair of hands to help him operate. You are in the operating theatre helping operate on your own father. You impress the surgeon with your calmness and dexterity. He offers to treat your father for free in return for your daily assistance in the operating theatre. 5 years later you are doing operations on your own. 5 years after that you arrive in the UK as a refugee, without being able to read a word, but with a detailed understanding of the human anatomy, and a wealth of medical experience.
That is one example. Another example is a family I knew who already had medically trained people amongst them when the war started in their homeland. A sibling group was orphaned, and taken in by an uncle who owned a hospital, and all of them started work there the same day. This group could in fact read and write to some extent, and their uncle taught them more, but he mostly taught them how to do operations, which was what was needed there and then, and gave them a skill they could earn a living by, whereas reading and writing , not so much.
And as to surgeons trained and qualified in the Western sense that you are talking about - yes, many many find themselves refugees. I have taken one in myself. He had trained in Europe 20 years earlier before returning to his home country. His identity and qualifications were later confirmed by staff at the university he had attended. he got refugee status, eventually, that was a long hard slog. he is now quite high up in the NHS.
Many doctors, surgeons and medical staff become refugees - they are in some cases known to be the targets of some regimes. The man who stayed with me had seen most of his faculty executed.