It seems that Peter was a common name among the Tierneys of County Cavan, and I don't know if any link will emerge, but please contact me if you discover one.Here goes...
I am descended from Peter Tierney born in Ireland circa 1835, in County Cavan. This I discovered in the 1901 census.
Peter married Judith (or Julia) Gaffney, also of Ireland, b 1836.
Birth dates are derived from the census.
Both settled Clifford in North Yorkshire, England, and I have the rest of the tree.
The next generation moved on to Knaresborough North Yorkshire, but there were clearly other Tierneys in Wetherby and surrounding areas.
I have so far been unable so far to make the link to Cavan.
There were Tierney's in Clifford (Nr Wetherby N. Yorks before the Great hunger) and there were strong links to St Edward King & Confessor Roman Catholic Church in Clifford and Boston Spa Area (North Yorkshire).
Rose Tierney was connected with the Convent of Sisters of Mercy in Clifford and Boston Spa Area and this is an invitation to anyone who may have a connection to post follow up.
As your ancestor's birth place and date are similar to mine, I wonder if there could be a connection.