Panther
This sort of situation can bring out the very worst in some parts of families - as well as the best - so it needs to be strictly controlled.
From what you say, your Nan is a vulnerable adult. She may have the onset of dementia or just simply be old but it looks as if she needs formal help. The last thing she deserves is angst in her later years, particularly with money. (She probably knows deep down that things are not right - otherwise why would she be phoning her bank and being agitated. She shouldn't have to: most bills should be able to be paid automatically by DD or CC (with some cash around for incidentals) and it would be far less hassle for your mom to supervise those on-line than to deal with what is currently happening.)
What Purple2012 says makes much sense - it's similar to what we did with my Mum and Dad; and they liked the regularity and safety of that method of operation.
You'll forgive me as well for mentioning this but does your family know what the situation is with your Nan's estate eg does she own her own home? I'm thinking about what might have to happen if she either has to go into care or when she dies. We had the 'luxury' of previously drawn up wills which couldn't be changed easily but you may not have that - at least as far as you know; and it's amazing nonetheless how much property/family heirlooms and mementos can go sideways when there's a vulture or two around. (I hate to use that word but that's what it is.)
I'd be speaking to the GP and to a solicitor as a matter of urgency. The situation will not improve from here on in and regularizing everyones' positions may be the only way for the family to move forward.
By the way, you can find out if your Nan has a proper Power of Attorney. Have a look round this guide and site.
\link{https://www.gov.uk/power-of-attorney\POA}
There are lots of other resources available online for you to check out as well.