Sorry in advance for long response and q's.
Like others have said this would fall under safeguarding a vulnerable adult.
Your sister needs an assessment under the Care Act. It should be your sister who is asked whether she would like this and if she refuses or your mum refuses access there are ways the local authority can intervene through using the Mental Capacity Act if necessary.
Do you know if your mum is deputy for health and welfare for your sister? It sounds as if she is not acting in your sister's best interests by refusing help or refusing to address the inevitable. It is something we see all the time when working with families in similar situations.
In cases where it is needed they can go to Court to ask for the powers to remove a person to a safe place such as a care home, obviously this is likely to be more distressing so they will need to evidence they have tried or risks would need to be severe enough for action to be taken without attempts first, and they'd need to evidence why they believe your mum isn't acting in your sisters best interests.
Social services would also likely call a professionals meeting / safeguarding meeting to discuss risks/develop a plan with GP / LD team and social care and to ensure agencies are sharing info.
if your sister became very unwell / was putting herself or others at risk the mental health act could be used.
I'd also be seeing if she has been under the care if the LD nursing team, like others have said they can provided support and education to your sister and mum and would work alongside a social worker. As well as helping to get the necessary health checks done for your sister.
It is likely the first thing offered would be a support package in the community, this could be a way to get support going in the home or for your sister to start going out with someone.
As your sister has PIP it is likely she'd need to contribute to the cost of social care. Do you know if your mum has a separate bank account for your sister? Do you know if she is using the money for your sister's needs/wants or keeping it herself or saving it up? Does your mum receive carers allowance?
Another thing would be for a carers assessment to be offered for your mum.
Have you spoken to a charity like Mencap for support/ advice?
There was a recent case where a woman with Downs's Syndrome died as a result of severe neglect, it is a tough read due to the nature of what happened but the serious case review highlights learning points/best practice, and you could mention this if you struggle to get anywhere with the local authority:
www.blackpoolsafeguarding.org.uk/assets/uploads/Adult%20V%20SAR%20Learning%20Brief%20final%20280323.pdf