Clearly you need to get out, to save yourself and your children. You've tried living with her and it's too hard - none of your siblings even bothered trying to care for her but you did.
Start looking for another house now. Don't tell your siblings or your mother - no need to do that until the house purchase is going through.
Is your mother capable of living on her own at the moment, without being in actual danger? If so, you could leave it until shortly before you move to ask social services for her to be assessed. If you think she will need care once you've moved, SS should be contacted earlier than that - would your siblings need to be informed about that?
Do social services need to be involved at all in the short term? Could you not simply arrange for private carers for her, until her condition deteriorates, paid for by her? Do you have Power of Attorney for her? If not, you might still have time to set it up if she still has some capacity.
(The Martin Lewis Money Show covered Power of Attorney this week, and the episode is on ITVx: https://www.itv.com/watch/the-martin-lewis-money-show-live/2a1827/2a1827a0169 )
I would suggest getting advice from Citizen's Advice or Age UK, as soon as possible, to clarify what you need to do to get appropriate care for her, at what point social services need to be involved, etc.
(There's lots of information and advice on the Age UK website about getting care: https://www.ageuk.org.uk/information-advice/care/ )
If necessary, could your doctor provide evidence of your mental health struggles with this situation, to provide justification for you moving out to save your health?
But please, please get out. You can't sacrifice yourself and your children when there are other options. You're lucky that you have the means to buy another home, so please do it.
It's not an easy situation but you do have a route out, so please take it. I wish you luck.