Since he died, my mother can keep her bank account because it already existed, but she can't open new accounts because she doesn't have the necessary ID
www.tsb.co.uk/current-accounts/faqs/identity/
Needs one piece of ID, which includes "Benefits/state pension notification letter" which she presumably has, and one proof of address which includes utility bills, council tax bills.
Either, but not both, can be replaced by an old-style paper driving licence. You can use the benefits letter for address if not using it for ID, and vice versa.
No need for photo ID.
So she can't put her money into a savings account and earn interest on it because she can't access one
See above. TSB is just the first bank that came to hand: the requirements are roughly the same for all of them.