Hey treacle,
I opened a student checking account with HSBC in the US. They call themselves the "world local bank".
While they are in the US and the UK, they are separate. With a debit card, I can access funds from the US account (with fees etc) - but I can't just go into a branch and withdraw money. I'm pretty sure that any account at any bank can be accessed from here with a debit card (and fees).
The proof of address is the sticking point, as I think they want your name on a utility bill. Students living in dorms must have some alternative, like a letter from the uni.
From what I gather, most people have pay flat deposits etc. with multiple ATM withdrawls.
Do I remember you saying that your parents are Brits? If you have relatives in country who are willing, you could wire them money which they could then wire directly to the account of a landlord, if that is your concern.
Best,
Holly