I haven't validated any of this but...
(also I'm assuming that you mean Bampton not Brampton)
...
The IGI has a christening of a Joseph Nicholson on 15 Dec 1826 in Bampton to William and Eleanor. Other children are William (1813), Frank (1818), John (1823), Margaret (1825), Rachel (1831).
I looked for the marriage of a William Nicholson to an Eleanor and found one to Eleanor Johnston (but in Barton not Bampton) on 23 May 1812. The IGI gives the age of William as 32 and Eleanor as 20 which implies that William was born around 1780.
It is possible that this was William's second marriage. There is a marriage of a William to an Esther Todhunter in Bampton 1804 and this is followed by 2 children: Mary (1806) and Isabella (1807). But this could be a different William.
I found the christening of a William Nicholson to a William and Rachel on 2 Jan 1780 in Bampton. Rachel is not that common a name at this time so I think someone previously assumed a connection between the Barton marriage and this Bampton christening and that is what the IGI shows.
The 1829 directory lists William Nicholson as a blacksmith in Bampton.
The 1787 constables census for Bampton has a household headed by William Nicholson (blacksmith). He has a wife, 2 sons and a daughter.
There are other Nicholsons on the 1829 directory and in the 1787 constables return.
I suggest that you check the parish registers for all the above.
