USA Citizen residing in the UK, paying tax in the USA on income paid in the US

My wife and I recently moved to the UK through my work. My wife (who is an American Citizen) is still being employed at the same job in the US, and is receiving her income in the US and paying tax within the US on her income. Does she need to apply for a UK tax number, and how is her income that she is paying tax in the US treated in the UK?

Hi,
Your wife will need to review the guidance on residence at RDR3  and take the residency tests, to determine her tax residency status in the 2023 to 2024 tax year.  
This will determine whether she needs to declare the overseas earnings in 2023 to 2024 on a self assessment tax return.  
If your wife is UK tax resident for 2024 to 2025, then she will be taxable in the UK on her Overseas employment income.  
As this income is taxable in the USA, she will need to claim a foreign tax credit in her self assessment tax return, so that she does not pay the tax twice:
Check how to register for Self Assessment
Thank you.