For any person with complex needs, making any change can feel enormous. It helps to chunk it down into small battles, but she probably needs assistance just prioritising what needs to be changed before she can actually move foward.
You seem to think that just dragging the child off to school, assuming one will take him, will fix things.
It won't.
It may in fact create a whole raft of other problems. If she is germaphobic (I presume you mean she experiences OCD) she might find it unbearable to have to deal with the situation of picking up and dropping off from school. There could be escalating complications from any intervention which she does not agree to.
The mother has needs that must be addressed before the children's situation can be changed.
As the children are not in danger, this is not a break down the door and change things scenario, this is a one brick at a time scenario and will take time. I very much doubt you are suitably trained to counsel her and help her improve their lives.
If you are genuinely concerned, contact your local social services, make it clear you want to remain anonymous and outline that your cousin needs resources and support, ask them to make an assessment and ask for advice from them, not mumsnet.