A bit late to join the party, but I am another descendant of John and Mary Bulger - I am a descendant of Mary Ann Bulger who married William Pulley and I am researching this part of the tree along with a few others.
I have the same info re John Bulger snr and his son Michael being convicts. I have dates of death of 8/8/1857, Bathurst for John and a burial of 1/10/1855 for Mary Bulger nee Feeley.
I believe they must be buried at Bathurst but they are not listed on the online cemetery site - however not all transcriptions are listed.
I believe that John's father was probably called Simon as there is a listing for a John Bulger died 1857 with Simon as the father on the NSW BDM - and he called one of his sons Simon.
Happy to communicate further - would like to swap the descendants of Mary Ann Bulger/William Pulley with the other lines if anyone is interested.