Civil registration of births in Ireland didn't start until 1864 so you won't find birth certificates for children born in Ireland before that date.
Civil registration of marriages also 1864 for all marriages but from 1845 onwards for non-Catholic marriages which include registry office ones.
This is why you probably haven't found anything in Irish Genealogy database yet. In both cases for earlier records you need to search for church registers. If Thomas' wife Mary was Catholic and the children were baptised as Catholic then you might find details in R.C. Parish registers-
https://registers.nli.ie/However, you might be better off checking all possible English records for clues first to narrow down locations. For example,
Have you followed wife Mary and all of the Irish-born children in census records to see if a more exact place of birth than Ireland is recorded? The 1911 census is the one that most often seems to include a place but worth checking all of them just in case.
Another possibility might be Army registrations but someone here will likely know more about those.
