I ran into the same dilemma of the bank wanting evidence of an address (e.g. bills sent to the address etc) and the real estate agent needing a bank account before I could rent a place.

In the end I went to HSBC's Passport Account. This is set up specifically for newcomers who don't have an address and other details that you might need to set up a bank account elsewhere. In order to set it up you need proof of identity such as a passport and some money to deposit in the form of cash or a bank draft. There are a couple of "catches" (that they make clear to you at the time) - you pay either an up front fee of around £60 or a monthly fee of £6 and you have to have the account for 12 months before you can "graduate" to a proper account. Otherwise there are no fees (i.e. withdrawals, deposits are free). You only have basic facilities - i.e. a cheque account with a debit card, with internet and phone banking. You can't get a credit card or overdraft on the passport account. I haven't found this to be a major issue - generally you can use your debit card where you would normally use a credit card (e.g. on the internet). There is the odd occasion where I haven't been able to use the debit card, so hanging onto my NZ credit card has been useful.

You can pick up your debit card and activate it a few days after setting up the account either from the branch or you can have it sent to a temporary address if you have one.