Thomas’s birth was long before the start of statutory birth registration in Ireland (1864). Normally in that situation you search church records for a baptisms. You haven’t said what denomination he was but I’ll guess Catholic.
There are 2 townlands called Tandragee in Armagh. There are no Nugents in Ballymore (which has one) but I do see Nugents in Lisnadill parish. The tithe applotment records in 1832 list a number including one farm held by Peter Nugent in Tandragee. That could be Thomas’s family.
http://www.irishgenealogyhub.com/armagh/tithe-applotments/lisnadill-parish.php
No Nugents in the townland by the time of Griffiths Valuation in 1864.
http://www.irishgenealogyhub.com/armagh/griffiths-valuation/lisnadill-parish.php
The tithes were a tax on land, so that was primarily farmers. Servants, labourers, people staying with relatives etc weren’t listed (because they had no land) so there might have been other Nugents around, just not listed.
No Nugents in Tandragee in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Lisnadill/Tandr…
Statutory birth registration didn’t start in Ireland till 1864. Prior to that you normally rely on church records. I assume the Nugents were Catholic. Unfortunately the parish records for that area (RC parish of Kilclooney) don’t start till 1844 so you are unlikely to find any record of Thomas’s birth, or of his relationship to Peter.
There are few other records that might include a reference to Thomas, so researching him in Ireland may prove hard going.
Possibly DNA testing may also be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).
