Hi Sarahjane1967
what a tricky situation your family is in
Do you live in the same area as your sister so that you can contact your mum's GP and consultants about this change of address and circumstances? Maybe you've already done that. Both certainly need to know, especially if your sister has found that your mum's behaviour has deteriorated - you don't say why your sister can no longer cope (and, of course, you don't have to).
I would contact Social Services as soon as you can and ask for immediate support - mention that your mum is a vulnerable adult and at risk as you must work and can't be with her at all times. Has there already been an assessment of her needs? Maybe your sister had a care package in place so you could contact the provider? If she is self funding, I would have thought that you could make/ arrangements yourself.
If your mum's behaviour has changed dramatically and suddenly, she may have a UTI, so definitely worth having that checked by her GP and get her on antibiotics asap.
best wishes