It depends on your state because my husband is not on his first child birth certificate with his ex and he has to pay child support and never did a paternity test and now has 50% custody we live in California
If you are in the U.K., the laws of your country will probably differ than in the United States. Some rights might extend until the child is of legal age, but those might be rights the child has to the parent, not rights the parent has to the child. Ask a solicitor or a legal-services agency or child-related agency.
If hes not on the birth certificate he has NO legal rights until a paternity test is done. Nor does he have to support the child. You can ask for a DNA test through the courts but Im not sure of the costs. It also depends on the age of the child. I think after 5 years the birth certificate is barred and there's not much they can do. Again Im not for sure about that but I think it's after 4 years in Texas they wont change anything.
If he's proven to be the father before the time limit he might have to pay back child support. My BIL has a 4 year old DD that he had no idea existed and the courts didnt make him pay back because he had no idea the child was his, but he does have to pay child support now for her.