Adoption didn't become legal until 1927?
Whilst it is correct that there was no statute creating adoption law until after the Adoption of Children Act, 1926, that did not make the adoption of children before that illegal, there were many adoptions under the Poor Law system, I would recommend Jenny Keating's book "A Child for Keeps: the History of Adoption in England, 1918-45".
However in answer to the original questions.
No, if the adoption was before 1927 the child's birth certificate would record the details at birth, what normally happened would be that only a short certificate would be obtained rather than a full certificate.
After the 1926 Act a full certificate could also be supplied showing the adoptive parents names (noted as such) and the date and name of the adoption court. Having said the be aware that informal child adoption continued well into the 1930s.
Cheers
Guy