Another thing you could try, 1841 census search for Edward, no surname, the right age, perhaps 3/4 in tormoham area, as the Bulley family seem to be there, and track them on to next census.Then check parents marriage, date would have to be after his birth, am guessing. Could be that he was born somewhere else, and brought to the area when he was an infant, so would not necessarily know where he was baptised.
I think he was illegitimate, perhaps brought up by mother and stepfather, or "adopted"by family, perhaps his mother died in childbirth, all sorts of scenarios are possible. It will take time, but you may find out in the end!! Have you checked wills of Bulley names in that area? Property transactions, that is where I found my missing ancestor, similar circumstances, no baptism under the name we were searching, baptised under fathers name, married under "fathers" name, then reverted to mothers name when baptising children.
In our case, I think both parents died, and children, 2 of them, brought up by fathers family. Only found out mothers name from an affidavit with sale documents relating to a property which was sold by his son. Good luck.