Lots of children in those days, and long after, were born at home so it wasn't uncommon for mother's to register such children without any paperwork in those days.
If mother listed but no father on birth or baptism this indicates an illegitimate birth. A foundling, if turned in to the authorities may have been fostered but it seems unlikely that someone finding baby on doorstep would be registering such a child- and certainly not without a note on at least birth registration.
Foundlings are sometimes mentioned in news reports so that's one avenue to search but it may be that the foundling story was to cover up illegitimacy.
Have you tried looking for information on the mother on birth certificate?