For the location of adoption records in England and Wales (from 1927), Scotland (from 1931) and Northern Ireland see http://www.familyrecords.gov.uk/.

For England and Wales the certificates themselves are held by the Office for National Statistics in Southport. Further details are given on the General Register Office website. The Office for National Statistics has a series of booklets you can request from them:

Access to birth records

The Adoption Contact Register

Information for adopted people and their relatives

Georgina Stafford's book, Where to Find Adoption Records, which was issued by the adoption agencies in 1993.