I've scanned replies here but not read them all.
I'm shocked that this was allowed to happen. If your sister has been living with your mum for years, and was latterly her carer, how can it be that no-one had a clear conversation with your mum and both your sisters, about future plans?
In Ireland (so I appreciate would very likely be different in the UK), someone in this situation could contest the will on the basis of having had a reasonable belief in this property being her home, and as such the will reflecting this, and the arguable sacrifice she made as your mum's carer (an earlier post notes the saving that was achieved by avoiding residential care). Indeed, there are specific provisions in law for these situations so that inheritance tax is avoided, due to the house being a principal private residence for your sister (inheritance tax thresholds are lifetime thresholds in Ireland, different to the UK and so can be very problematic in terms of paying tax due).
That's all moot as it doesn't apply to this situation.
But I think the wrong questions are being asked. Firstly, clear legal advice is needed, secondly an honest conversation, approached sensitively, with your DSis.
But for anyone else in this situation, for the love of pete, do not wait until death to sort it out - have clear conversations early on, and if this situation is planned to arise, be very clear what will happen re: house sale, re-housing of sibling in situ etc.
I can understand she is being difficult, and that's not fair to you, but I have a great deal of sympathy for her. What a huge upheaval ahead for her.