Hey sorry for the delay in responding! I was coincidentally enough too hung up working in the hospital (and my clinic) to check my reddit notifications!

The answer to your question is most "yes", though you chose an interesting time to ask about (so maybe the answer is "it depends"). Unlike France and Germany, the UK (and the US, which outside of Louisiana had largely just adopted the English model) did not traditionally have funded hospital positions, and when they did (such as part of a medical school), the funding was relatively meager such that physicians would essentially be forced to have their own private practice to make a living. What did this look like for a hospitalized patient? By the early 20th century, if you were wealthy enough to pay a physician's fees and you needed to be hospitalized (say, for salvarsan therapy for syphilis), you would be admitted to a private pay ward, and your own doctor would round on you every so often. You would pay his fee, as well as the hospital fee. If you were poor, you would be admitted to an indigent ward, and a staff doctor would round on you -- but that same doctor would likely see his own patients in the afternoon.

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Now the answer is largely "it depends" because in the early 20th century, both the US and Britain started to adopt the German medical university model (in the US, we call it, in a very American way, the Hopkins model). One of the essentially components of this training model was "house officers" -- trainee physicians and surgeons would exclusively worked in the hospital, or hospital-based clinics, and therefore did not see private patients. They were ostensibly staffed by attending physicians, but in this period largely worked independently. In the US (and therefore much of the rest of the world), we call them "residents" since they used to live in the hospital. So these doctors, for the period of their residency (as little as one year, but sometimes up to five) would exclusively care for patients in the hospital and associated clinics, and not see any private patients.

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Unfortunately, I'm not as familiar with the timing in the UK for this model to be fully adopted; in the US, it gradually spread among larger universities in the late 19th century, leading to the Flexner report in 1910 and this essentially becoming the only model of post-graduate medical education.

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Hope that helps! Let me know if you have any follow up questions!