If you know their name after adoption and their year of birth, you can check the GRO Adopted Childrens Index at any of the 7 GRO host sites, get the official reference and apply for a certificate.
The Library of Birmingham
Bridgend Local and Family History Centre
City of Westminster Archives Centre
Manchester Central Library
Newcastle City Library
Plymouth Central Library and
The British Library
http://www.gro.gov.uk/gro/content/certificates/most_customers_want_to_know.asp#AdoptionsI can check the index for you next time I visit if you can't get to any of the host sites. Please send me a pm with the name and year of birth.
The Adoption certificate will not contain the name at birth.