How long does it take to include child’s name in birth certificate?

My daughter was born in February 2024 and we submitted a form for her name inclusion in the Puzhuthivakkam zonal office in March. Even after 10 months, her name has not been included, though I can see other children’s names being updated on the list. We tried to register online again (we don’t live in Chennai) in January, and Google says 3-4 days to update, but still nothing! Has anybody here faced this type of issue? I’ve raised a complaint but it’s very difficult to even get her an Aadhar without a birth certificate with her name.