I've gotten a bank account now, at HSBC. It was fortunate that I went there, as being an existing customer allowed me to open the account with less proof than I'd have otherwise needed.
Less proof consisted of my passport and tenancy agreement, instead of that as well as a letter from my company and a utility bill.
Now I have to wait for my bank card. I tried to get a phone with my tenancy agreement and passport, and that didn't work. They told me I'd have no problems when I returned with my debit card.
Utility bills seem to be a very common way to prove credit-worthiness here in the UK. However, they only send them once a quarter. That means you could theoretically wait almost three months before you can purchase anything of significant value. That seems … less than effective for the purchaser, though I'm guessing they have far lower costs for charge-backs here.