Proof of address is the challenge! There are very few documents that meet the requirements. You can thank the hostile environment for that, if you aren't in the country legally they don't want you able to bank. I had more luck with Barclays than HSBC, as i had my US HSBC statements sent to my UK address and that worked for them but not HSBC. Council tax or a utility bill are the other most common proofs, but each bank will have a list of all valid options. You may have to push for a branch manager if it's not council tax or utility bill though. It will likely take a couple weeks minimum to get the UK account set up. Keep trying and good luck!