Griffith Valuation 1856 shows no Timothy Cooney in 1856 so he was probably too young to have his own rented land.
There is however:
Thomas Cooney, Parish of Kilbeacanty, Townland of Gortacarnaun, Poor Law Union of Gort, tenant of James Kelly
#2 house, outbuilding and land 41. acres 3 roods 11 perches
#3 1/5 share of bog land 10.1.32
There maybe other Cooney's in the PLU of Gort, but in other Parish's. The Union would of covered about a 12 mi. radius and would have any parish's, town/villages and townlands.
Looking for Kilbeacant Parish came across
There were only 2 Richardson, one wealthy in land.
Kilbeacanty & Peterswell Parish in the Diocese of Galway,
Kilmacdugh & Kilfenora. The church there is St. Columbawww.galwaydiocese.ie
You also lucky that Kilmacduagh records start in 1853 with marriages only, so you should be able to find a marriage
in the 60's. Birth records don't seem to exist but
by this time they would be showing up in the Civil Registration records (1864+) both on an Irish History Foundation not for profit but fee paying
or free www.familysearch.org
Children were often taken up by relatives or put in the Workhouse or working in church facilities then orphans were often then shipped off to the colonies for cheap labour. They may have been lucky and sent to relatives in the US.
Timothy Cooney died Jan-March 1893 age 28 estimate year of birth at 1865 *Newcastle, Galway.
Total speculation: Perhaps this is the oldest son? took care of the children till he died and then the other's went to America, it would depend on the date they arrived in the US.
Margaret Cooney died Jan-March 1873 Galway, age 42, estimate year of birth as 1831 based on age given at death
There are Cooney's in the Galway telephone book, 1 from Gort and Richardson's as well, 1 in Newcastle* addresses given are sufficient for mail delivery.
If they were still in Ireland by 1901/11 you should be able to find them in the census:www.census.nationalibrary.ie
Without names I can't help you in that search.