Hi Adam
There is an entry in the birth index in 1934 for Eileen M Havill, is this your grandmother?
If it is, you can but a copy of the birth certificate from the General Register Office, this will show whether she was adopted or not as there will be a note made on the right hand side of the certificate.
If she was adopted, there should also be an entry for her in the GRO adopted childrens index, but you would need to know the surname she was adopted as to find any possible entries.
She married in 1960, but using the surname Havill. Was she adopted by someone within the family so her surname didn't change?
Dawn