On your theory of the adoption BC. I know that an adoption BC will be generated at the time of adoption and will show the adopted parents as if they were the birth parents.
Would such a BC have original signatures (doctor, mother and registrar)? And the dates that the people signed the certificate, would they be backdated to the date of birth?
Yes. Here is an example of mine, created 6 years (1967) after I was born.
Note it contains the Doctor's signature as if he signed this document.