Aha, that's fair. Judging by the site it seems you'll have to take care to demonstrate that you're not shifting UK derived profits; but either way you'll want to chat up an accountant.
In my jurisdiction, to my knowledge and experience, they're considerably more lax.
The US company will pay 20% UK Corporation Tax on (worldwide) profits and need to fill out CT600 each year.
You will probably be able to pay lower taxes in the US as a result of double taxation treaties, but it's going to be a very tedious task to organise.