Ha, we're having this problem (we live in the UK already, but our lease is expiring in a few months at the same time that my work contract ends and my fiance graduates from uni, so we'll be potentially without jobs and a house at the same time.) I would focus on getting somewhere to live first. How much do you have in savings? If it's a sizable amount, that might help with landlords. If it's not, then staying in a hostel/serviced apartment/B&B for a few weeks might be what you have to do while you look for a job. If you don't have a job, I'd stay away from estate agencies- private landlords are usually more flexible though.
As far as banks go, I know that there is at least one that will theoretically let you open a bank account with just your passport (it's either Lloyds TSB or Barclays, check their websites). But usually you need proof of address, so I'd do flat-bank-job (or really, flat first and job and bank hopefully at about the same time.)