Hi,
If you are a US citzen, then I'd expect it'd be much easier for you to open a business bank account with any of the larger USA banks. You'd then operate the account as 'if' you were in the USA and get a merchant ID to accept card payments etc electronically. Once you've been paid in the USA, you could then use a currency exchange specialist to transfer the funds to your UK bank account at better exchange rates than you'd find on the UK high street/banks etc.
If you are a UK citizen with minimal USA 'financial' presence, then without a USA registered business it'd be much harder to get a business bank account opened over there. However, I think some of the larger banks there in the USA do indeed offer to open a USA Dollar account for a business registered here/out of the USA.
When I opened up my personal USA bank accounts in branch on my last US visit, I did ask about getting a business account. I was told that by operating the personal ones now and building up an 'internal' rating with them, then when it came time for me to 'expand' my USA operations to include more payments from customers, that I could then apply for a business account.
The banking industry has changed somewhat over the last 2 years since I opened those personal accounts, but I'd say the general fundamentals are still the same. It'd be worthwhile you speaking to some of the larger USA banks to see if things could be easier right now with you being a business owner with USA based clients wishing to pay you in USD
If you do find more out, please do post up here the information you get?
Cheers, DtM! West London & Slough UK!