We're going through it right now with my mother-in-law, the only (positive) difference being that she was in a care home to begin with, so we haven't had to deal with the actual care side. Like your granny, she's been 'losing it' for some time and it was a spell in hospital that really pushed her over the edge. She took an accidental overdose of her drugs, plus she had an infection as well. She self-medicated until that point and so is naturally very upset that her carers have taken over the administration of her meds. We've also had to remove her phone because she called 999 twice and told the police that her carers had stolen her meds.
People with dementia act just that - demented. There is often seemingly no logic to what they do and say and they have the memory span of a fly sometimes. What they say can be quite upsetting at times too (my MIL, for instance, is forever singing the praises of the somewhat dastardly but now dead male friend with whom she grew up as though they were siblings, but never mentions her lovely late husband).
Try to humour her, but only when it's safe. I find that MIL is helped by being shown old family photographs, even though the memories make her sad sometimes. I was able to have some of her memoirs and photos published into a book, and I've noticed it's always marked at a different page when I visit, so I know she's looking at it.
You should not try to cope with this alone, and neither should your Mum. It's important that you both show your support for each other as you seem to be doing. Neither should either of you feel guilty if the situation makes you feel angry or in need of a break. Caring for someone with dementia is very stressful and those who do it professionally will fully understand you.
You don't say how old you are, but I'm guessing you're still at school. This website is Oxfordshire based, but it offers some helpful information: