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.
How can my child apply for indefinite leave to remain if my wife does not have this status?
My family applied for the tier 1 investor visa four years ago. I was the principal applicant of the case. Last year my wife and I divorced and she returned to our home country. Recently, I learned that in order for a dependent child to qualify for indefinite leave to remain (ILR), both parents have to obtain the ILR status first. I have full custody of my son and I already got the consent of my wife to apply for ILR for him if we are able to. In our case, can my son still apply for ILR even if his mother left the UK?