She doesn't need a bank account to top up a PAYG mobile. You can still get vouchers in all the shops, post office, anywhere that does PayPoint for bills etc. Can pay for vouchers in cash or card. Putting the voucher on the phone is very easy too. Call 150 for EE, 2345 for Vodafone or 4444 for O2 then enter the voucher number.
OP's mother doesn't have to worry about previous council tax. She's liable for it all from the date of grandmother death as she's the person living there. Same as tenants pay the council tax on a rental and not the homeowner. She needs to claim the 25% single person reduction for being only one adult living there and also Council Tax Reduction from the council as a benefit, because of low income.
Previous council tax was down to the grandmother. If granny was doing fraud by claiming the 25% discount when there were two adults living there, well nothing can be done about that now because you can't prosecute or fine a dead person.
OP's mother wasn't even a lodger but even if she was, lodger's don't pay council tax, the homeowner does because the homeowner still lives there. It doesn't get split between the people living there only one person pays it.
I guess if you jointly own property with a spouse you're both "jointly and severally" (think that's the term) liable for it. Meaning the council can chase either of you through the courts for the whole amount, if it isn't paid. This isn't the case for OP's mum because she never owned the property.
She can register online for the electoral roll, it's easy.
If she doesn't get social housing for whatever reason and ends up needing private rental the council will help her find a landlord that accepts benefits and help her with a deposit if necessary, so long as she engages with the homelessness procedure.
She'll probably only be able to afford a room in a shared house or HMO, unless she lives in an area where rents aren't too far over the Local Housing Allowance (housing benefit for private housing).
She'll have a regular income for rental, it'll be Universal Credit. Unless she's stupid enough to get herself sanctioned for non-compliance.
If she doesn't want to a) get a job or b) jump through the hoops to get benefits, she's free to live on the street. It's not actually illegal to be homeless I don't think. I never heard of anyone being arrested purely for that, anyway.
If she does end up on the street, she can still claim UC when her period of being sanctioned by being kicked off it has ended, but obviously you don't get the housing elements because you're not paying rent. You don't need an address to claim benefits, you give the local jobcentre as your address and check there for DWP post regularly (you can't use them as an address for other post I don't think).
The bank can mark your account with a fraud risk warning meaning the bank won't send out bank statements to your last address (which will remain on your bank account, because you can't have a bank account without an address) but you can still use the account the same as usual with the card and in branch, probably online/telephone banking too.
If you're homeless and don't have a bank account, the jobcentre will issue you a Giro cheque for you to cash at the post office and hope you don't get mugged. If one chooses it is possible to live on the streets, though I can't imagine that's a pleasant life. You can get a GP still for healthcare.