Hi NYstacey
I called up Amex to speak about this some time back, I seem to recall that as you're in the US, you'd contact Amex over there to begin the process. Once you get yourself settled here which probably means once you have an address, they'll 'transfer' the account over to Amex UK. By doing this, it keeps your history with them intact, and they'll then submit the monthly 'report' on you to the various credit agencies here.
Amex allow this between the various countries it operates in, so not just limited to US to UK. I asked about my card, and it can be done UK to US, UK to Canada. I didn't ask about other countries.
As per Jim's preference, you can of course still continue to use your US cards here in the UK, but I think if you are living here permanently, you'd be in breach of your T&C's with the credit card company - might be worth looking into that small print though.
I rang up MBNA as well, I can't remember exactly what they said, i seem to recall they said they wouldn't transfer the account as such, but would allow me to open an account there and show I've shut my UK one down .. I think....
Hope this helps ! Cheers! DtM! West London & Slough UK!