The Birth Date is a rough estimate. See the text for details.
Biography
John was a Friend (Quaker)
Research Notes
No birth or death record located in the Irish Quaker records.
John Wiley is briefly mentioned in the 1740 will of his grandmother, Elizabeth Watson nee Thompson (Thompson-26699).
There are Quaker marriage records for his father's two marriages, but no birth records for any of the children.
When granting approval in 1729 for his father's second marriage, the Quakers notice that ‘he had made provision for a child by a previous wife’.
In 1750 the Quaker monthly meeting for Dublin granted a John Wiley a removal certificate addressed to Friends in Pennsylvania or elsewhere in America, but it cannot be pinned down to this John Wiley.
Then there is a John Wiley son of Robert Wiley who married Anne Barnes in Mountmellick, Queen's County, Ireland in 1758, who is more likely to have been him.
Other than that I cannot find any sources for him.
The John Wiley mentioned in the will is very unlikely to have been born in the north of Ireland as both his parents were from King's County (as was his father's second wife) and there is no record of them having moved north.
Sources
Ireland, Society Of Friends (Quaker) Family Lists, image, FindMyPast (https://search.findmypast.co.uk/record?id=IRE%2FQUAKER%2FQM1M-3%2F0314&parentid=IRE%2FQUAKER%2FMARR%2F7197B : accessed 25 October 2019), marriage of John Wyly son of Robert Wyly and Ann Barnes daughter of Jonathan Barnes in Mountmellick, Queen's County on 25d 10mo (Oct) 1758; citing Leinster QM Marriages 1716-1775, Religious Society Of Friends In Ireland Archives.
Ireland, Society Of Friends (Quaker) Migration Records, image, FindMyPast (https://www.findmypast.co.uk/transcript?id=IRE/QUAKER/MIG/1072 : accessed 18 Apr 2021), removal certificate for John Wyly to visit Philadelphia and other places in America on business, dated 5d 4mo (Jun) 1750; citing Dublin MM certificates of removals. 1682-1784.
Source: S9 Ancestry Family Trees (Online publication - Provo, UT, USA: Ancestry.com. Original data: Family Tree files submitted by Ancestry members.) m. Original data: Family Tree files submitted by Ancestry members. TID 0 Footnote Ancestry Family Trees (Online publication - Provo, UT, USA: Ancestry.com. Original data: Family Tree files submitted by Ancestry members.) ShortFootnote Ancestry Family Trees Bibliography Ancestry Family Trees. Online publication - Provo, UT, USA: Ancestry.com. Original data: Family Tree files submitted by Ancestry members. Repository: #R1 Ancestry Family Tree Page Ancestry Family Tree http://trees.ancestry.com/pt/AMTCitationRedir.aspx?tid=69911325&pid=4108
Repository: R1 Ancestry.com
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Wyly-61 and Wyly-33 appear to represent the same person because: John Wyly from the first marriage survived as he is mentioned in his maternal grandmother's will so there is unlikely to have been a separate John Wyly from the second marriage.
Wylie-902 and Wyly-33 appear to represent the same person because: Two people with the same first name born to the same parents and both living to maturity are unlikely.
I have plainly made a mistake with this profile, having the date of birth well before the parents' marriage. Sorry. It probably needs to be merged with one of the other children.
Robert Wyly and Ruth Watson lived first in Offally and then in Kildare, both in the south of Ireland. I don't think that they would have had a child in Down in the north.
Alan
Alan