This is a fun question!

First - gentry and aristocracy are not the same. Aristocracy are a level up from gentry, basically members of titled families with some limited bleedover into people with significant political power and/or very high levels of multigenerational land wealth. Gentry would be financially indepdendent from their land-related income (not "trade") but less wealthy, less powerful, and not noble.

Doctor was a perfectly reasonable profession for a member of the gentry in 1920 and for some time before then - most doctors of that period had such a background (Joseph Bell, Edward Jenner, Alexander Fleming, Freud, Henry Littlejohn, even Treasure Island's fictional Dr Livesy). By the 1920s medicine was scientific(ish) and respectable - though plenty of quacks were still in practice the "sawbones" approach was dying in the 18th century and dead by the 20th. Few people who weren't from such a background (albeit in some cases barely!) could afford medical school or found a practice.

For an aristocrat - somewhat less likely because it looks rather like trade. Aristocrats were under stern societal pressure not to appear to be dragged down to needing to involve themselves in any kind of trade or business and to be supported solely by income from rents and perhaps government stock. In The Picture of Dorian Gray (1890) owning land that had coalmines on it was an embarrassing "taint of industry"; in The Way We Live Now down-on-their-luck aristocrats are embarrassed to be reduced to lending their name to an investment scam. Aristocrats had a few legitimate ways of spending their time

• Diverting themselves (yachting,hunting, shooting, travelling and so on)
• Government
• War
• Enthusiastic pursuit of any branch of art or science as an amateur.
• Possibly being a barrister (definitely not a solicitor!)
• Possibly being a churchman (if a second or later son)

Even managing your own estate too enthusiastically was a bit infra dig (unless presented as an enthusiastic amateur agricultural scientist dragging his estates into modern systems of management). You were supposed to have an agent for that sort of thing.

An aristocrat, especially a second son or someone on the boundary between aristocrat and gentry, might in the 19th century have practiced as an amateur doctor (probably with a tight specialism) such that the practice fell into the "enthusiastic pursuit" category mentioned above. Conceivably they might have been doctors in practice for free, as a vocation, helping the poor - but such a practice and lifestyle would have been most eccentric. Anything approaching a commercial medical practice would have been a big no.

However, by the 1920s the position was less pronounced than 100 years earlier and doctors (especially leaders in their fields, or trained but non-practicing doctors) could quite conceivably be aristocrats. By 1935 in Biggles Hits The Trail it was realistic for Lord Maltenham to be a medical doctor and doubtless there are other literary and real examples. They would probably have been keen to present themselves as enthusiastic amateurs (taking few cases based on interest or personal relationships, being driven by a personal interest in a challenging and specific specialisation in respect of which they were particularly talented, or simply not practising after qualifying) but they could easily have been medical doctors.

It's worth noting that a lot of the reason for this is that practicing as a doctor when you're an aristocrat, then or now, is just a bit weird. Obviously some people have a strong vocation and will practice as GPs/family doctors regardless of their level of wealth. But put yourself in the place of an aristocrat (again - then or now) and ask both why you'd do it and how you'd end up wanting to:

• You are hugely rich - even a top consultant's salary will have a negligible impact, sub 1% over your entire career, on your wealth.
• For the same reason, even if you are inclined to "do good" there are thousands of other ways to do so that have a broader, deeper impact
• I won't go into detail beyond that except to say that thinking over the ambitions, shortcomings, idealism and consideration of other opportunities that everybody goes through in deciding what to do with their lives is not the same if you grow up in an aristocratic home with aristocratic friends, education and decisions open to you. Basically you grow up exercising power and enjoying doing what you want to, and you will tend to keep doing that forever!

Read e.g. the biography (or even wiki) of Lord Cardigan and you'll get more of a sense of how much wealth and opportunity for diversion and lionization, and how little respect for non-aristocrats, was in the psyche of an Earl. He was an incredibly brave person and at times he gave enormous support to his men; but he rarely (until towards the end of his life) had any conception that he might be wrong about anything and was quite capable of leaving his dead and dying soldiers on a battlefield to eat lobster and drink champagne on his yacht, or letting his regiment's horses die for lack of fodder out of pride.

A final note - there are not and never have been a lot of aristocrats in the UK. There are just over 800 hereditary peers. Perhaps five to ten thousand people are aristocrats if you draw a wide net. Those people still live today much as they always have, they are just less visible because of a larger overall population, a tendency to hold larger proportions of their dynastic wealth in financial instruments, easy international travel, preferences for privacy in the face of modern media, and universal suffrage. Most Guards officers still spend more in the mess than they receive in pay each month. Almost every prime minister and chancellor and CGS went to a major public school. Etc.

I have a few past answers that relate to this question, though anyone is welcome to write a fresh one that directly responds to it!

What did it mean to be a gentleman in Britain in the Victorian era? How did it change over time? (/u/kingconani also has an answer there that's even more thorough.)

Why did the ultra wealthy start having nominal occupations instead of just being aristocrats?

What did landed people do all day?

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