An Update:
Stopped in the Royal Bank of Scotland soon after arrival to confirm what they'd need to open an account. Was told a passport and a bill or a copy of our lease.
Got a lease on a place May 4. Phoned RBS and did an application over the phone for an account, and was instructed to take the lease and passport to a local branch to finalize the account. May 5 we walked to the bank, only to be told that they would not accept the lease as proof of residence because it's with a private landlord they didn't know and that they only accept leases from entities like the Council. (Of course, it turns out that the letting company is a subsidiary of the Housing Authority, they just were not familiar with it.)
So, phoned to get electric and gas turned on, and the following week received correspondence from the power company with the requisite details. Phoned the bank, was told that would be sufficient, so made an appointment to walk back to the bank yesterday to get the account set up. And now have an account, debit card (etc.) to arrive by mail any time now. So, start to finish it took me about 10 calendar days to get an account. Apparently they pay 3% back to you on your household bills, so by doing direct debit on the rent, council tax, Sky internet, etc., I'll actually make a few bucks... 'scuse me... Pounds on having my money with them.
Works for me!
Was getting ready to apply to be covered at a nearby GP when the computer virus thing hit - probably best to wait a bit for them to deal with that rather than showing up when they can't turn a computer on!