Hi Becki
Legal adoption in England & Wales commenced in 1927, earlier adoptions therefore won't be recorded anywhere.
The birth should be registered in the name that the mother was using at the time of the birth.
I have seen more than one instance where the mothers surname on the birth cert is not the one you would have thought it was, most probably done to conceal the situation. Sometimes the grandmothers surname was used

There is a GRO adopted Childrens Register but it's not available online and only available on microfiche at the 7 host sites around the UK,
Birmingham Central Library
Bridgend Local and Family History Centre
City of Westminster Archives Centre
Manchester City Library
Newcastle City Library
Plymouth Central Library and
The British Library
If you can't get to one of the locations send me the a pm with the adopted name and I'll have a look next time I'm at the archives.
There is no way of linking an original birth entry with an entry in the adopted childrens register unless you know both names.
Dawn