Our daughter is in exactly the same situation having moved here in November last year, and is still working for her US firm now while they try to sort out paying her out of the UK.
As a UK resident she has to pay UK tax on her worldwide income and as a US citizen she also has to pay tax on her worldwide income but the UK-US tax treaty states that the UK is the primary taxation authority and tax credits will avoid double taxation. So, when she filed her US taxes for 2022 she claimed an accrued foreign tax credit for the UK taxes she will pay on her US income in November and December, 2022. She will do the same for the current 2023 tax year when she files her US return next year. She will be filing her UK return for 2022 later this year when she know exactly how much UK tax is to be paid. The difference between accrued tax she claimed and the actual tax will be reconciled on her US 2023 return.