From what I remember, you don't need much money to actually open the account. Some people have trouble opening accounts, as KFdancer alluded to, but I had no issues whatsoever-- I had armed myself with a letter from my Uni, but didn't need it, as they were happy to accept just my US Passport (though this was >3 years ago).
But like KFdancer suggests, it could take a while, so you'll need to have enough money to support yourself in the meantime. I'm not sure who you bank with in the US right now, but you could check to see if they have any agreements with UK-based banks so you can take money out when/as needed (For example, Bank of America customers can take money out at Barclays cashpoints with no additional fees).
Now that I'm over here, I use TransferWise to send money back/forth directly into my US/UK accounts. It's normally quite quick (2-3 days), and has best exchange rate combined with the lowest fees I've found so far.
If you're interested, I can give you a free trial by referring you (so just PM me if interested). At least this way, you wouldn't have to carry around a load of cash until after you've found a place to put it.