‘ The social worker would have referred it to the legal team as your sister's actions are a safeguarding issue. With no capacity to transfer your sister may well have commited fraud.’
Unfortunately there are solicitors around who do not follow best practice when it comes to determining if a client, particularly if they have dementia, has capacity to make a will or give instructions.
I suggest that you take prompt legal advice to determine if your mother can give you some form of power of attorney - given her degree of dementia - & then approach her dementia consultant & the social worker to work out the best solution. This is so that you can formally advocate for your mum about her wellbeing & any financial matters. Your sister has ruled herself out of the game given her false declaration & the safeguarding alert.
Now is the time to step up to resolve these matters. Get informed & work with the social services to sort things out for your mum so that she can settle in her new place.
Also get legal advice about how the existing will can be revoked - be cautious in your approach because if your mum did not have capacity then, she won’t have it now. It may be that a will made before she developed dementia can stand.
What is important is that your mother is safe & well in residential care that suits her & if you & your mum want to do this, you have some form of attorney for both her health/wellbeing & financial matters so that you can advocate on her behalf.
It may be that the safeguarding review leads to the legal team raising her case with the Court of Protection. This is a significant step to protect your mother & would likely take place after attempts to resolve matters by the local social services.
Get legal advice, get informed.