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 these India List posts
Adoption was a private process, perhaps arranged by a family member, as mentioned in this India List post 
The churches appear to have had a role in adoptions or placements.
- 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 India List post regards a child, whose parents were not married to each other, born at the Lady Dufferin Hospital in Lucknow in 1943. At 10 days old he was removed from the hospital by the Minister of the Methodist Episcopal Church at Lalbagh and placed in the care of a couple who happened to be in Lucknow at the time involved in church work. The family subsequently returned to Scotland. He was never legally adopted.