As far as I know, the tax part is handled on the writers end independently.

The companies that I worked for made it clear that they do not do the withholding tax thing but that they would assist the writer in cases when tax payments would be necessary.
The inclusion of the form in the documents sent to you were probably part of that. No need to worry, you don't have to pay for taxes if you are physically working in the United States.
I don't believe that there is double taxation for online workers because you basically still work in your home country, regardless of the currency you are paid in.
You just have to be sure that you set aside your tax money for payment of personal income tax every year.
Keep track of your salary via the Paypal statements or whatever documents are used as proof of salary so that you can present the required paperwork with your tax papers when the taxman cometh in the UK.
The US taxman doesn't have anything to do with it.