In your case scenario, your child can apply for indefinite leave to remain at the same time with you, provided that you have since divorced sole responsibility for your child's upbringing. However, you will need to provide evidence to prove that you have been involved in your child's upbringing since the divorce, not the child's mother. A legal document granting sole custody is not sufficient. Applications involving sole responsibility are always not straightforward and it is important to prepare as much valid evidence as you can. I strongly advise you to instruct a professional legal advisor to assist you with the application.
