If one bank doesn't work, try another. One does not need any work permit, residence permit or any other kind of permit to open a bank account. What the bank needs is sufficient proof for the "Know your customer" and the money laundering rules, as well as some proof of address. The latter is virtually any bill except a mobile phone bill.
HSBC is actually easier than many other institutions. A reference from a foreign branch will do, and HSBC is worldwide. Building societies, including Nationwide (which is almost a bank) tend to be relatively easy. It's been a few years but I recall they gave me some trouble over my national ID card (equivalent to a passport for entry to the UK), probably because they see them so rarely. I know that my son and his wife (who had just arrived from abroad) opened an account with Abbey (now Santander) with just passports and perhaps a council tax bill in my (not his) name.