You can do 20 hours per week work (full time in term time). I see no reason why that can't be for a US employer. The tax aspects can't be answered on this section of the board, but I have certainly not seen any requirement to pay tax in the UK.
I would say that you would need to submit a UK tax return each year, which would disclose the money you are paid and the tax you pay in the US (if you would prefer to do that), and that this would satisfy the Home Office on renewal, along with evidence in the form of a letter from your employer that you are only working the proscribed hours and what your hourly rate is.
Vicky