In the 1980's I requested a birth record for my grandpa from the U.K. post office in London. The researcher couldn't find his name or anything like it for that year or for several years before or after. My cousin's son discovered a letter from Grandpa's dad indicating that he and his wife adopted him.
The legal process for adoption the UK wasn't introduced until 1926/27. Before that adoptions were informal agreements and did not need to be legally registered.
https://www.jstor.org/stable/30009455?seq=1 will give you some background information.
The answer at https://genealogy.stackexchange.com/a/2801/6485 may also give you some ideas of how to investigate.