Do you mean that the baby's birth mother did not register the baby? That would be illegal. An adoption registration if this was after 1928 would have to be linked to a natural birth registration.
I know a couple who did this in the 60s, the single mother went into hospital with their name and they took the baby from the hospital and registered it as theirs, they were childless and called the little girl Storm and they all lived happily ever after, the birth mother knew how to get in touch but never did.No money changed hands either, friends of both sides put them in touch.
The baby was given to the adopted parents at 1 week old. It was a private adoption and there is no record of a registration in the local record office !
I don;t think you can go down the usual genealogy route if you do not know the name the child was given after being handed to the adoptive parents, not unless you know who they were, this is the place to start:
http://www.norcap.org.uk/