Working for a US company remotely

I have a question that Google doesn't seem to have a definitive answer for - is it legal to move to the UK (from the US) while continuing my present job, working remotely? My job allows me to work from anywhere, and while I'm more than happy to pay the appropriate Visa fees, I cannot determine which Visa to apply for. 

Does anyone have any experience with this? I would also maintain my current US bank account, if that is needed.

Yes, kind of.

You need a work visa in which you would need a sponsor.  Does your company have offices in the UK?  If so, would they be willing to sponsor a visa for you?  It's not as simple as live in the UK and work for a US company.  You can do that if you have the appropriate visa.