Hi Keith - welcome to Rootschat

I had a bit of a trawl through various things at the Northants Record Office and found out a bit more about William:
On his marriage entry to Elizabeth Green it states he was married by license.
On the license it gives his occupation as weaver of Wellingborough.
In the Wellingborough parish records there was a record of William Bryant apprenticed to Thomas Williamson, woolcomber & weaver at Wellingborough in 1752.
There was also a militia paper stating that William Bryant of Wellingborough, weaver would serve as a substitute to Robert Burgess of Brixworth who was chosen by lot in 1783.
There were the following baptisms at Wellingborough:
William son of William Bryant born 11 Oct 1736, bapt 28 Jan 1736/7
William son of William Bryant bapt 11 May 1740Elizabeth daughter of William Bryant bapt 23 July 1738
Winifred age 13, Benjamin age 8, Sarah age 4 the children of William Bryant all bapt 30 April 1756
It looks as though the first William must have died prior to 1740 when William 2 was named after him.
Obviously wives weren't considered important enough in childbirth to get a mention at thier offspring's baptisms

The Greens were a bit more prevalent in the baptism register:
John Green, plaisterer age 21, bapt 18 Oct 1725
Mary d John Green, plaisterer, bapt 21 March 1726/7
John s John Green, plaisterer, bapt 21 April 1728
Martha d John Green, plaisterer, bapt 25 Sept 1729
Hester d John Green, plaisterer, bapt 11 Jan 1731/2
Anne d John Green, plaisterer, bapt 8 April 1733
Francis s John Green, plaisterer, bapt 28 March 1736
Elizabeth d of John Green bapt 19 March 1737/8Katherine d John Green bapt 11 Aug 1739
Lucy d John Green bapt 30 May 1742
Esther d John Green bapt 6 Jan 1752
Again the mother is not mentioned so it is impossible to tell whether they are all children of the same parents - Esther may be the child of a different John Green or possibly from a second marriage to account for the 10 year age gap.
Where does Rushden come into it (as in the title of your post?)
Hope that helps a bit

regards
byron