Hello, I am Keith, one of the experts on Just Answer, and pleased to be able to help you with your question.
If you reside in the UK for more than 183 days in any one tax year then you are liable to UK taxation on your world wide income. However, any taxes paid in the States will, under the Double Taxation Treaty between the UK and the USA (and indeed many individual States in the Union a well), which precludes the same income stream being taxed in both jurisdictions, are allowed as a tax credit against any UK tax liability on the same income stream. The Treaty does not, however, protect you from differences in rates of taxation. You will have to make a self assessment tax return at the conclusion of each tax year [5 April] to HMRC. On receipt HMRC will advise you of any tax and NI due.
PAYE is a method of deducting Income Tax and National Insurance contributions introduced in the early 1940s; my Mother was responsible for its implementation within the British Broadcasting Corporation. It is used by employers and would appear inappropriate in your situation as you US employer pays you stateside.
I do hope that you have found my reply of assistance.
Thank you for your support.