I wouldn’t necessarily say Northern Ireland was mostly Church of Ireland. There were plenty of Roman Catholics too, and Presbyterianism is/was more common than the Church of Ireland. In my opinion though, your family was probably RC. (98% of the Donnelly households in Tyrone were RC according to the 1901 census).
Anne Donnelly’s death cert says she was a farmers widow. Does that tie in with information on Australian certificates, ie that Patrick was a farmer?
I looked in Griffiths Valuation for 1860. Griffiths should list most farmers. There was only 1 Donnelly farm in Ballybray townland that year and it was occupied by Bernard Donnelly. Given her age, my feeling therefore is that this Ann Donnelly was probably married to Bernard.
http://www.askaboutireland.ieThe revaluation records show that Bernard ceased to be the tenant in 1863 when he was replaced by James (his son perhaps?). Probably indicates Bernard had died. James in turn was replaced by Eliza Quinn in 1875. There are no Donnelly farms listed in the townland after that point. Going by her death certificate, Anne was evidently still living there so possibly lodging with a relative. A married daughter perhaps?
By 1901 the farm was evidently headed by Alice Quinn:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Donaghmore/Ballybray/1739092/Now the tithe applotment records for 1826 do list a Patrick Donnally in Ballybray (plus 2 other Donnally farms) so that is an interesting piece of information:
http://www.cotyroneireland.com/tithe/t-a-donaghmore.htmlBallybray is in the RC parish of Donaghmore. It has no records earlier than 1837. However there is a note against those records in the PRONI catalogue to say that earlier baptisms and marriages may be found in the adjacent parish of Killeeshil (whose records start in 1816). Since this Anne probably married in the 1820s or 1830s, you might want to go through those early records looking for a marriage of a Donnelly from Ballybray to a Kenny, to see if this family can be ruled in or out.
The RC parish records are on-line free on the National Library site:
http://registers.nli.ie/parishesI notice you give Anne’s age on the death cert as 83. And her date of death as 31.3.1887. The GRONI indexes give her age as 79, and date of death as 5.3.1887. Is there a mistake there? I ask about the age because if she was 84, she was pushing it to be the mother of a child born in 1851/53. Though ages on death certificates are often out by a few years, being just the informants best guess.
I suppose the next question to ask is how reliable is the information that your Anne died in 1887 and that it was in the Dungannon area. If the year is out or the location is out, you may need to widen your search a bit. As Hallmark has said, there are other deaths around that time of women whose age fits your information (10.1.1886 aged 73, and 12.1.1888 aged 63 being two). You might need to have a look at them. Ann(e) Donnelly is a very common name in Tyrone and so this may not be easy.