In my experience the best exchange rate is likely to be found by withdrawing cash from a British ATM using an American debit card.
However you can also get hit with fees this way so you need to make sure you use the right card. Visa/MasterCard charge your bank 1% to do currency conversion, and most banks (quite fairly, IMO) pass this on to you. Many banks also add their own charge on top - normally 2% - for a total 3% charge. Many also charge you a separate fee for using an ATM that doesn't belong to them - these can be $1-$5 per withdrawal. If you are using one of the big banks like Citi, Chase, Bank of America, you are almost certainly getting overcharged in fees.
On the other hand, there are a few (mostly smaller) US banks out there that will actually absorb the 1% charge from the card companies so you get the "interbank" exchange rate with no further charges, which is as good as it gets.
I use Charles Schwab, which is one such bank. I've had nothing but good experiences with them the past 3 years. If you can get an account, highly recommended. They also refund any charges made by the bank who owns the ATM. There may be other banks out there that will also eat the 1% charge.
The only limit would be the amount of cash you can take out of the ATM every day (probably in the $1000-$2500 range).