Your sequence has flaw;
1. If any birth is not found registered, the Registrar shall issue a Non Availability Certificate (NAC is issued under The Registration of Marriages, Births & Deaths Act 1969 State Amendments included).
1.1 First approach the MO and ask for a BC form. You will get it free of cost. Now fill up the form which is a single page Form - Fill the child's name and parents' name in the relevant sections, fill no. of copies section as 1, leave the Name of hospital (where born) as blank if child born at widow's home, fill the section "Place of Birth" with the address of the widow from where your clients have adopted the child (donot put the name of the widow but just her address), submit the form along with a copy of adoption deed. Pay Fees.
2. The MO checks at the concerned address for the child availability. Since the child was not born there, it would issue a Non Availability Certificate (NAC) once you collect free of cost Form, fill it and submit along with requisite fees.
3. The MO would ask you to collect the NAC on a due date.
4. On the due date go to MO, collect the NAC and use the same to file a petition at the District Magistrate's office.
5. Court then will issue an Order for the MO to generate a BC (birth certificate).
6. Using the Court Order of the ld. Court again apply at MO following 1.1.
7. Now MO will generate a BC (birth certificate) with the parents' name in a case where no prior Birth Certificate was found on record and child was adopted and registered adoption deed also exists.
7.1 CARA which is Central Agency has already issued strict guidelines that the name of adoptive parents shall appear as “parents” and the term “Adoptive Parents” or “Adopted Child” shall not be used in the birth certificate in reference to facts of your query. [Chapter VI, Rule 82, sub-clause 2 and 4 is helpful reference for your facts].
Hope your query is clarified?