sounds really stressful .
My thoughts are to involve the support of your school nurse. Perhaps in the first instance, they could have a chat with parent about distinguishing between illness and anxiety responses (not easy, but the parent should have some pointers).
The nurse will be able to liaise with the GP in case of support needed for mum.
Have you got access to early help, perhaps for respite support over the summer break?
I do find that when you involve an external agency for support, parents listen more. it also sometimes helps them to understand how important it is schools find solutions to these issues.