atticus,
This is Walter Campbell’s birth certificate which tells us his parents were Francis Campbell & Margaret Weir.
https://www.irishgenealogy.ie/files/civil/birth_returns/births_1868/03446/2265233.pdf
Here’s Francis & Margaret’s marriage certificate in 1859. Both had been married before, she to a Leonard. Francis was an army pensioner who had served with the 15th Regiment. You may find some details of his military history on Ancestry or findmypast., Other more detailed records may be available at the National Archives in Kew, London. Normally a personal visit is required to view them. (Or get a London based researcher to do it for you). There are also Regimental diaries which show where the 15th Regiment served, which may help give more detail on where Francis served in the army. There was no conscription in Ireland and so Francis’ decision to join the army was his alone.
https://www.irishgenealogy.ie/files/civil/marriage_returns/marriages_1859/09576/5472374.pdf
There are trees on Ancestry which have this family. One shows Francis (b 1820) having the following children by his first wife:
Francis 1843 in Thurles, Co Tipperary
Catherine 21.2.1850 in Brecon, Wales
Mary in May 1850 in Newry,
James 1852 in Mullingar Co. Westmeath
So the first marriage, if in Ireland, was before the start of statutory marriage registration (April 1845) and may be hard to trace without a location. There’s obviously an error in some of the information because it’s quite unusual for a woman to give birth to 2 children just 3 months apart - Catherine & Mary 1850).
Walter Campbell b 1868 doesn’t appear in the 1901 Irish census which suggests he had left Ireland by that year. I don’t see any marriage for him in Ireland.
James Campbell married Sarah Jane McMoran in Newry in 1882. They had 1 child born in Ireland (in 1883) and then were in New York by 1887.
https://www.irishgenealogy.ie/files/civil/marriage_returns/marriages_1882/10966/8008375.pdf
James died in Pasadena, Ca in 1931 according to the Bradley family tree.
There are no censuses for Newry in the years 1861 - 1891.
Hopefully some of this helps.