I wonder what happened at the actual service though, would both parents be expected to be there, together with godparents? I wonder if a vicar would question it if just the mother turned up?
As the second baptism took place a few years after Edward was born Hannah may have implied that she was a widow.
Edward's birth and baptisms happened during the Napoleonic Wars. Many fathers would have been absent when their children were born and baptised. Some would have been in army or navy. Civilians were recruited or conscripted into militias and moved around Britain and Ireland.
Even if a father was at home, unless the baptism was on a Sunday he would probably have been working and unable to attend.
At the time when a baby was baptised soon after birth, the mother would not attend; she stayed home, recovering from the birth. In places where it was customary for a woman to return to her mother's home to give birth, maternal grandparents would take the baby to church for baptism.