Hi,
Sorry to hear about your situation. Because of the nature of your issue, have you not been in contact with support, either governmental or your local council? Perhaps, social services? Due to such a delay in your education, I would've thought someone would be supporting you through this. I've got a friend who has suffered family problems too, and his social/youth worker sorted out his GCSEs for him, and I don't think he had to pay to sit them. Maybe look into contacting some local services that can help you with this.
Best of luck.