Hi,
Does your father have a copy of his adoption certificate as this could have his birth name on it as well as his mother's name? If not, there's a useful link here that might help him obtain it
https://www.gov.uk/adoption-recordsIf he does know his birth name, or at least his mother's maiden name (and therefore likely his birth surname) you should be able to find his birth record and order a certificate from the GRO.
I have recently ordered a birth cert for a colleague's husband who was adopted - he knew his birth name from his adoption cert so it was a simple process.
Adoption records are famously hard to come by though if you are trying to obtain them for another person rather than yourself - is your father living or deceased, if you don't mind me asking?
If you want to share any more info I'm willing to see if I can help shed any light but can't promise anything! :O)
Thanks,
Becca