Supposedly the seller of a car should be the one filling out V5C/2 ‘new keeper supplement’ of the log book with the purchaser's name and address. Not sure if they are meant to check or not.
For the rest, opening accounts without photo ID is pretty tricky, here's what you need according to one random bank's website if you don't have one of passport, EU(~) driving license or national ID card, or firearms license:
Valid older style UK driving licence (no photo)
HMRC documentation (PAYE Coding Notice/Tax Notification/Self-Assessment/Statement of Account/NI contributions bill) issued in the last 3 months or valid for the current tax year. We can’t accept P45 or P60 forms
Notification letter from Benefits Agency/Local Authority confirming your right to benefits (Department for Work and Pensions including Jobcentre Plus, Benefits Agency or Veterans Agency), dated within the last 12 months.