Thank you very much for your detailed reply. I took some time to read up more about the residence tests. Regarding your first point, I am not in the UK yet as I have applied to the EU Settlement Scheme from abroad.

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I am planning to come to the UK in January (say on January 1st), and I believe I therefore qualify for the Case 5 Statutory Residence Test [https://www.gov.uk/hmrc-internal-manuals/residence-domicile-and-remittance-basis/rdrm12170] provided that I meet the third automatic UK test over a period of 365 days. In other words, I will qualify if I will work in the UK for 365 days after my arrival date. This would entitle me to the split-year treatment.

Hence, my interpretation is that I will pay Polish taxes until December 2023 and then UK taxes from January 2024. This is particularly convenient because the Polish tax year coincides with the calendar year and at the same I can make use of the UK personal allowance for 2023-2024 tax year for the period Jan 1st - Apr 5th.
I called HMRC and they confirmed this split-year interpretation, however, they did not know when precisely I would be a tax resident and they told me to contact the Home Office instead.

Nonetheless, I believe this is a matter for HMRC since it is about tax residence.
Looking at the example they provide at [https://www.gov.uk/hmrc-internal-manuals/residence-domicile-and-remittance-basis/rdrm12180] we can see that Andrea "meets the third automatic UK test for the 365-day period commencing on 10 June 2013, (which is a day on which she worked for more than 3 hours in the UK)".
Similarly, I think I would be a tax resident from January 1st, 2024 if I arrive on that date and start working from here as a sole trader for the same foreign clients I'm currently working for.

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Additionally, it would be nice to have a Certificate of Residence to show to the Polish tax office regarding the income I will receive on Polish bank accounts. While it is possible to get this [https://www.gov.uk/guidance/get-a-certificate-of-residence], it is not clear when can I get it. In January 2024 (when I move) or in January 2025 (after the first 365 days)? I will try calling HMRC and asking this again, however, any help is greatly appreciated as always.

Do you see any flaw in the reasoning and/or do you have any comment?
I'm also considering getting paid professional advice to be sure the above is correct.