Re: Thomas Smith of Northamptonshire, England
The only safe way to do genealogy is to work backwards from what you know into the unknown.
You appear to know that Joseph Smith married Louisa Warner in Buckinghamshire in 1866. In subsequent censuses Joseph is shown as being born in Wicken Northants c1842/3 and Louisa inLeckhampstead Bucks (the two parishes are only a mile apart). So far so good. The problem comes with the 1851 and 1861 censuses where Joseph isn't living with his parents, just with his grandparents George and Mary Read in 1851.
Joseph Smith son of Thomas and Eliza Smith was baptised at Wicken on 16 Oct 1842 (per IGI).
Contrary to what you say, the only Thomas Smith/Eliza(beth) marriage in the GRO index in Northants is in the June quarter 1842 Potterspury registration district, which includes Wicken, where the other party is Eliza Read. This ties in very nicely with the 1851 census where Joseph is living with his Read grandparents.
And in 1841 in Wicken was George and Mary Read and daughter Eliza 22. It all fits so far. Now to find what happened to Thomas and Eliza after the baptism of Joseph in 1842. As you've already found, they were living in Lillingstone Daynell, Bucks, which whilst being in a different county is only a mile across the county boundary from Wicken, Northants. Thomas was 34 b Hanslope which is in Buckinghamshire (5 miles from Wicken) and Elisa 30 or 31 b Potterspury (the age is partially obscured) plus 3 children from age 5 downwards. So it looks as though the 1842 marriage works timewise. In 1861 they'd moved to Buckingham with an increased family, where they were still living in 1871 and 1881. When and where they died I don't know.
So having followed Joseph and his parents through censuses back to 1841 I don't understand why you sent yourself on a wild goose chase with the Northampton Thomas and Elizabeth and sons Eli and Joseph. Nothing about them, other than their names and the county, ties in with your Thomas and Eliza, even if you allow for Eliza and Elizabeth being the same. But the Northampton couple were married probably before 1835 with no evidence at all that Elizabeth might have been Read. In fact I think she may have been Elizabeth Garratt who married Thomas Smith in Oct 1833 in Northampton. But it's immaterial as they're nothing to do with your family.
All that remains is to prove that the 1842 marriage is the right one by purchasing the marriage cert from the GRO site, which should show fathers' names. You might then stand a chance of finding Thomas in Hanslope, although I can't see that the parish register is available online. There is a Thomas Smith of the right age, 20 ie 20-24, in Hanslope in 1841 living in the same household as Mary Smith 45, but you'll more than this before you can be sure that this is the right Thomas.