Well, that was easier than I thought it would be.
Background - my partner was transfered to the UK by her company, on a Tier 2 visa. I'm here as her dependent.
We walked into the Reading HSBC branch, armed with passports and a letter from her company stating our temporary address and that she was employed by them in the UK. That was all fine for setting her up with an account, but to make it a joint account with me the bank woman had to get her manager's approval - which took about five minutes. The whole process took an hour or so. Tons of forms, tons of information, but once we decided which account type we were qualified for and wanted it was rather simple. Now we're just waiting for the cards to come in the mail - 3-5 business days, she said.