I had to check that I was on the legal board when I saw the responses - people do type up a lot of guff.
Grandparents go to court on a regular basis and very regularly get contact with their grandchildren, whether they have met them before or not. I am not saying OP, that your mother WILL get contact, but I am saying that it is not the impossibility that so many of the non-lawyers on this thread have asserted to you.
You are not obligated to respond to solicitors letters, they have no more legal force than your mother writing to you directly. But, if you ignore them, then you are very likely to find that an application has been issued in court.
Yes, this will first of all deal with the grandmother's permission, which she needs to achieve before the substantive application is reviewed, but this is almost always done for child arrangements at the FHDRA (first hearing), it will be the first matter discussed, and I have rarely seen such permission refused unless there are extremely compelling reasons to deny it. This is the point, OP, where you will need to raise your mother's conduct, why you don't have contact with her and why you do not want her to have contact with the children. If your reasons are compelling enough, the court may refuse permission - but it's unlikely.
What is most likely to happen is that permission will be given and the issues you raise re abusive conduct will be investigated further by way of a s7 welfare report whereby CAFCASS or social services if you've had recent involvement. This report will investigate the allegations and do background checks and make recommendations to the court as to what the court should order. It may be indirect contact - grandmother allowed to send letters/cards a couple of times a year, and you may be asked to send photos back. Or it could be supervised contact in a contact centre, or similar. It will depend entirely on the outcome of the court's investigations and the court's decision as to what is in the best interests of the child.
To be clear, I am not saying your mother in particular will get permission, or that she will get a favourable order, but I am saying that this application type is not rare, and it is not rare for grandparents to get contact with grandchildren, even when they do not have a current established relationship.