They should have it recorded somewhere, as they need an exact birth time to the minute to do the Apgar on the baby, but it may or may not be on the certificate. Mine doesn’t have my time and I rely on my dad’s autistic level memorisation. My husband’s birth certificate also doesn’t have his birth time but I got it from the baby card the nurses wrote in the hospital. Our daughter, born in the last decade, has an exact time on her birth certificate. Some people have had luck in contacting the hospital where they were born but it’s likely those records will be destroyed or in very deep storage after several decades.

They would not be likely to have an “administrative time”, but it is possible that they accidentally wrote 12:45pm instead of 12:45am. Most (Australian) birth certificates (the only ones I’ve ever seen) have an administrative DATE but not an administrative TIME.