You say that during the conversation between your father and the solicitor, he agreed that your mother was unfit and wrote her out the will in the knowledge that you would take care of her. I think you mean that he wrote her out of the will. However, as the will was never executed, it’s academic because under the rules of intestacy your mother would inherit the majority of the estate anyway subject to the intestacy rules.
What the solicitor is doing is refusing to distribute the estate to your mother on the grounds that your mother does not have mental capacity. What you normally need therefore is a letter from your GP or your mother’s specialist or some other medical profession which confirms that she only has a communication and speech problem and not a mental health or understanding problem and that she is fully aware of what’s going on around her and in total control of her faculties. If the solicitor will not accept that letter from your mother’s GP, then I would ask the solicitor who she will accept a letter from. If the letter from the GP didn’t go into enough detail, then it’s easy enough to get one with more detail!
There is no reason why this cannot be sorted relatively easily. However, as this is an intestacy, there is no reason why you should have the solicitor dealing with it just because the solicitor dealt with the will. You could deal with the application for the grant of letters of administration yourself or instruct another firm of solicitors to do it provided the solicitor has not already started the process which you say he has not. Why not just find another solicitor?