The best source of names and land come from the Tithe Applotments of 1820-1839. This is now used as a census substitute. www.genealogy.nationalarchives.ie
There are no Neaton's listed in Tipperary but there is one in Offaly, Michael Neaton, Ballyboy, Kings/Offaly. Offaly & North Tipperary do share a border and parish's and townlands can overlap and marriages usually tok place in the bride's parish.
Next is 1846-1864 the Griffith Valuation (also of land)www.askaboutireland.ie
(you can see who lived where, how much land they rented/leased, who their neighbours and landlord were. Also important other than the Parish is the Poor Law Union that would cover a 12 mi. radius of the town named for the PLU.
The only Neaton in 1852 was named Mary Neaton renting from
Daniel Kennedy Parish Dorrah, Townland Derrinsallow
Poor Law Union Parsonstown also known as Birr
10Aa house, outbuildings and land 15 acres 18 perches
10B land 10 acres 2 roods and 19 perches
10Ab Mary Neaton is the landlord for Mary Moylan - and rents her a house
next of course is the 1901/11 which is late for your family.
Tipperary North and South Riding have records on an Irish History Foundation. Perhaps this is where you found your baptism records. I notice RC Birr-Carrig records only go back as far as 1838 and Dorrha only as far as 1886 for the Church of Ireland.
So you have done pretty well getting the information you have. There are no listings for the Neaton's in the Tipperary or Offaly telephone book.