Sorry to hear that your relationship with your mum has broken down, and that she is not well.
My husband and I looked into this earlier this year, and as far as we could understand, legally grandparents have no presumed rights. thus, mediation or a court order would have to be fought for.
My advice to you would be to contact your local citizens advice bureau on where you stand.
And if you haven't already done so, you might want to post this question in the Law section!!
Also, hopefully this will be a catalyst for your mum to get the help she needs, and you can start to mend your relationship in time!
Good luck
ERx