Do you happen to know whether the US has taken professional advice about how to proceed? Either directly, or through a local subsidiary? If not I think they really need to do that, because there are potentially a number of tax issues here.
For starters, it is possible that them having you as a UK-based employee might possibly cause them to be liable to UK Corporation Tax on the profits deemed to be caused by their UK sales.
As regards yourself, and on the assumption that your work will be in the UK, you will be liable for UK tax, but questions remain about how PAYE will be operated, and how NI contributions will get paid.
Of course if they had a UK-based subsidiary the operation of PAYE and NI would be easy and standard, but I am absolutely not necessarily saying a UK subsidiary is the best way to go. All this needs to be thoroughly looked into.