Each bank is free to determine their own list of acceptable documents (as long as they can be seen to be complying with the Money Laundering Regulations when doing so). For example, Barclays will accept "an HMRC tax notification dated within 12 months" in place of a passport or driving licence as proof of identity:
https://www.barclays.co.uk/personal/current-accounts/what-do-i-need-to-open-a-bank-account/
Further, if a person applying for an account doesn't have any of the documents listed, they're advised to "please call us on 0345 7345 345 to discuss the other options available".
https://www.barclays.co.uk/help/accounts/opening/what-id/
It's likely that other banks will have similar ways of proving one's ID available to new customers.
It's also possible that, as your wife is already named on a joint account with a bank, that bank might be prepared to accept that she's already proved her ID to them anyway.