There is a record of John Bradley’s marriage to Bridget McWilliams on 20.2.1855 in Ballinascreen parish. They were cousins (3rd degree of consanguinity).
https://registers.nli.ie/registers/vtls000632987#page/151/mode/1up
I searched for death for John & Bridget Bradley in Ireland. I found this for Bridget in 1885 showing she was a widow then:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1885/06281/4799750.pdf
Informant was her daughter Margaret.
I don’t see a death for John Bradley that definitely fits. I assume this is the family in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Londonderry/Bancran/Boncran_Glebe/1543159/
The Valuation revision records on the PRONI website show John Bradley on plot 59a in Bancran Glebe. It changed from John to Bridget in 1880, and to reps of Bridget in 1890, then to Laurence in 1897. So that points to John dying around 1880.
I don’t know for certain which school the family would have attended but there was a school in Bancran so that seems the most likely. It’s records don’t appear to have survived. (The old school appears to be a B & B now). PRONI have attendance records for 3 in the parish of Ballinascreen: Moneyeena National, Sixtown National and St Eugene’s aka Carnamoney National. The records are not on-line and so a personal visit is required to view them. (Or you might get a researcher to do that for you).
The nearest chapel looks to be Straw, and it has a graveyard, so I’d search there first for family graves. The parochial office or the Ballinascreen Historical Society might also be able to help.
https://ballinascreenhistoricalsociety.com
Bridget’s will is on the PRONI wills site. She left the farm to Laurence and asked her executors to give the rest of the family what they judged to be their respective share.
The Will of Bridget Bradley late of Bancran County Londonderry Widow who died 17 July 1885 at same place was proved at Londonderry by James Reid of Bancran Farmer one of the Executors.
Other probate abstracts for Bradleys from Bancran Glebe:
Probate of the Will of James Bradley late of Bancran County Londonderry Farmer who died 3 April 1901 granted at Londonderry to Charles M'Kenna Schoolmaster.
Probate of the Will of George Bradley late of Bancran, Co. Londonderry, who died 20 June 1919, granted at Londonderry 11 February 1920 to Dan Kelly, Farmer. Effects: £24 0s 0d.
Bradley Mary of Bancran Draperstown county Londonderry spinster died 10 July 1931 Administration W/A (limited) Londonderry 28 June to Lawrence Bradley farmer. Effects £10. Limited Grant.
Bradley Rose Ann of Bancran county Londonderry died 18 December 1945 at the Mater Hospital Belfast Administration Belfast 8 March to Thomas Bradley farmer the husband. Effects £104 5s. 8d.
Bradley James of Bancran Draperstown county Londonderry farmer died 23 November 1949 at Gransha Mental Hospital Londonderry Administration Londonderry 20 September to Brigid Bradley spinster. Effects £143 15s.
Bradley Mary of Bancran Draperstown county Londonderry spinster died 10 July 1931 Administration W/A (limited) Londonderry 28 June to Lawrence Bradley farmer. Effects £10. Limited Grant.
The above probate files are not on-line but should be in PRONI. They will copy them for a fee or you can go in person and view them free.