Doing US business abroad in the UK
I presume you are mostly concerned about the taxation part of your question. Since the focus of this website is primarily incorporation questions unfortunately I would not be able to answer that and I would recommend you to consult an accountant. In short, if you continue to file your personal taxes in the US the same should apply to your business (especially since its a sole proprietorship).
Other than that I don't see anything that you would need to do to continue doing your US business from abroad, unless you are required to register locally in the UK (but that is really outside the scope of this website).