Adoption
It appears there was no formal legal regulation of the adoption of children in British India, the government was not involved in the adoption process and accordingly there are no government records.
In fact there are no known sources of adoption records, apart from the fact that baptismal records may occasionally show original name and adopted name, as indicated in this India List post
Adoption was a private process, perhaps arranged by a family member, as mentioned in this India List post
The churches may also have arranged adoptions, at least the Roman Catholic church. The subject of this India List post was born in the Eden Maternity Home, Calcutta in 1917. She was then taken to St Vincent’s Home Kidderpore (Calcutta),run by a Roman Catholic order, from where she was adopted at the age of six weeks.
This further India List thread discusses various aspects.