Hi,
If you will continue to be paid by your US employer, while working in the UK, you will need to declare your foreign income in a Self Assessment tax return.
As you will no doubt need to declare your income to the IRS, you should claim a foreign tax credit for any tax paid in the UK on this income.
As a UK resident, the UK would be entitled to tax this income first.
You would have to contact the National Insurance helpline for advise on any National Insurance contributions payable.
National Insurance: general enquiries
Thank you.