The tithe applotment records list farmers in Ireland. There are 2 Nathaniel Smiths in Teeshan in 1825 (senior and junior so probably father & son).
There’s also a John McCaugheon nearby in Leymore. That shares a common border with Teeshan. The two properties were probably a few hundred yards apart. Both townlands are just on the northern edge of Ballymena.
1 Nathaniel Smyth in Teeshan in the 1901 census:
Here are some others Smyth/Smiths:
In 1901 Teeshan had 46 houses and a total population of 187.
No McCaughan families in Leymore by 1862 so possibly they had died.
Here’s a Nathaniel Smyth getting married in 1850, aged 35. He lived in Teeshan, so might be a candidate for the father of Nathaniel b 1842 – 1845.
They married in Ballymena Register Office, so they may have been Brethren.
If you are looking for the baptism of Nathaniel Smyth c 1842 – 1845 you would need to search the local church records. Family seem likely to be Presbyterian or perhaps Reformed Presbyterian. The records are unlikely to be available on-line.
If Reformed Presbyterian (aka Covenanters) they probably attended Cullybacky RP church. If Presbyterian you have a wide choice. There are about 5 Presbyterian churches in Ballymena and there’s the Cunningham Memorial in Cullybackey. Try all of those. There’s copies of the records in PRONI. Not all have records back to the time of Matilda’s birth but most have some for the 1840s. If the family were Brethren, there are no baptism records in the public domain.
There’s a couple of graves in Kirkinriola graveyard (Ballymena) for Smyth of Teeshan but I don’t see any for Nathaniel. But here’s a possible for a Nathaniel’s wife:
ERECTED BY Nathaniel Smyth of Tyshan IN MEMORY OF JANE his beloved wife who departed this life 6th October 1852 aged 64 years.
Here lies The body of Janet Smith Daugt [er] to Nathaniel Smit[h] of Tyshan who depart[ed] this life December the 20. 1807 aged one yea[r] six months
I didn’t find any McCaughan graves.