Just an update on this.
After a bit of digging, I have come to the conclusion that Elizabeth Hunter (1824-1912) is the mother of Thomas William Hunter. I believe she had Thomas out of wedlock which would make it almost impossible to locate his father.
Hopefully, someone can find some sort of evidence to prove a father for him, maybe a Newspaper Article or some sort of another record? (I do not hold hope though).
The reason why I came to this conclusion is that Elizabeth Hunter, after her marriage to William Coates on 18 Mar 1852 in Eastrington, Yorkshire, England, she can be found in the following Census Records.
1861 Census; FOUND, BUT UNABLE TO LOCATE PIECE, FOLIO, AND PAGE NUMBER
1871 Census; (Piece 4750, Folio 49, Page number 15)
1881 Census; (Piece 4723, Folio 56, Page number 23)
1891 Census; (Piece 3892, Folio 30)
1901 Census; (Piece 444, Folio 20, Page number 6)
1911 Census; (Piece 28406)
It is the 1881 Census that provides a clue to my question. Frank Hunter, the Grandson of the head of household William Coates/Elizabeth Hunter. Frank in this census is 4 years old, which would give his birth around 1877.
Now if we go back to Thomas William Hunter's children, Frank Hunter (1876) is missing from their household in the 1881 Census which can be located here (Piece 4447, Folio 38, Page number 23).
I believe that Frank Hunter is living with his grandmother Elizabeth here which ultimately makes Elizabeth the mother of Thomas William Hunter.
That is my belief anyway. If anyone can see any flaws in my speculation then I will be more than happy to revisit this.
