Domicile is not (generally) relevant for self-employed individuals.
The important tests are residence and where the work is done. Under the US/UK social security agreement the key test is ordinary residence for the self-employed.
You need also to bear in mind that just because someone is regrded for US purposes as an independent contractor does not mean that for the purposes of English employment law (and hence UK tax law) that they are not actually employed.
You need to look at the contractural relationship to find out if this is a contract of service (ie employed) or contract for services (eg one that permits him to work when and where he wants, use other subcontractors etc) (ie self-employed).
There are key employment law differences plus extremely different income tax, corporate tax, social security, US withholding, VAT and transfer pricing consequences in both countries.
Sorry if this note sounds scary, but tax is complex at times!