This page explains how a child whose parents have given up and subsequently resumed British citizenship can meet the requirements for registration as a British citizen.
A child will have an entitlement to be registered as a British citizen under section 3(1) of the British Nationality Act 1981 if:
British citizenship gained in your own right (not by descent through your parents or grandparents). This type of citizenship can be passed on to your own children. (See also Descent.)