I’m not sure from what you’ve said if you are looking for your grandfather’s birth certificate and also looking for his father?
Was your grandfather old enough to have been mentioned on the 1911 census (1921 due for release next year on Find My Past)? He may not have given his age correctly on his marriage certificate - I’ve come across that, so search a few years either side. Do you have his death certificate, which should give a date and place of birth, though that would only be as accurate as what he’d told the person registering the death?
As for his father, very difficult unless for instance your grandfather was given a middle name which was the surname of his father (not uncommon), or if your grandfather was born in a workhouse which might possibly have some useful records though they would probably be closed for 100 years. I have illegitimate ancestors but have never managed to find the fathers.
I have subscriptions to Ancestry and FMP and would be happy to have a look if you give me some details.