It deepends when you were born for one thing, since 1972 you will have been given a surname , before that just first names and the surname used was firstly the registered father and secondly the unmarried mother.
BUT: in all cases, if the parents are not married, the father has to be present at the time of registartion or within a short time after for his name to be included on the registration.
You could not possibly have been given a new birth certificate with altered details, the only way would have been an adoption certificate. there was no perjury, only mis-information that only DNA testing can now resolve.
You can use any name that appears on your birth certificate without changing it legally, though that only costs 30 quid anyway.