Allow me to value add;
1. Laws says if mother has been awarded custody and there is no as such condition got allowed by natural father then child's last name change is her decision.
2. In reference to context if natural father is said to be not providing any child support r/w has no meaningful interaction with minor then he is not in a position to object to child's last name change. Putting what I just said to test in reference to emerging situation (question) all the lady is suggested to do is to approach District Judge Court in ordinary child jurisdiction along with 'foster father" i.e. her second legal married husband and file papers for 'legal adoption of the child'. The District Court after “conducting enquiry” will wave off the pre-condition of "seeking consent" of child's natural father. This 'enquiry' is nothing but reading down True Certified Copy in original annexed with adoption petition which is lady's divorce judgment mentioning who has the physical custody of the child and any hint on alimony / maintenance / visitation rights / any pre-condition of no surname change till child attains majority etc. etc.
3. I differ to opinion of Mr. Sumitra (para 2) Guardian is appointed by Court and it is settled Law that neither a divorced mother nor a divorced father having custody allotted respectively can appoint a guardian on h/er own.
4. Birth Certificate containing biological father's name will remain the same and it can not be changed / modified. For all future purposes including ration card name inclusion, property share, investments, nominations, instruments deposits/ withdrawal, passport, driving license, PAN Card, Adhar Card, State Tel. connection, Domestic Gas connection, election card, Xth. Class Certificate and College degrees etc. valid adoption deed / decree is way ahead if foster father's surname is to be given to child.