I've been digging into this again, and here are some notes of what I know so far, using this thread and stuff I've found:
A Joseph Yellowley married Barbara at an unknown time. They had two children: William (bap. 20 June 1779) and John (bap. 8 July 1787) in Thockrington.
William married Frances Elliott in Bellingham on 1 February 1801. They had two children: Joseph (bap. 6 Dec 1801) and Mary (bap. 28 July 1803) in Thockrington. Frances died in 1805 aged 26, and was buried in Chollerton on 16 April.
William married Jane at an unknown time. They had two children: Thomas (born 9 March 1806, baptised 14 Oct 1807) and Barbara (born 5 Mar 1808, bap. 14 Dec 1809) in Thockrington.
A William Yellowley married Margaret Gustard in Hebburn on 20 July 1811. Their first child, James, was born on 11 April 1812 and baptised on 20 Dec 1812, but was listed as William's third son. The Thockrington William did have two older sons (Joseph b. 1801 and Thomas b. 1806) so this makes it more likely that the Thockrington William Yellowley and the Hebburn one are the same person. The Hebburn William Yellowley died in 1828 and age given as 48, so a birth in 1779 would make reasonable sense.
I can find no marriage record for Joseph Yellowley and Barbara, but I have found a likely death record for Barbara. It states she was born in 1756 and was buried on 18 Nov 1832 in Chollerton. A 1756 birth year would fit with William's birth, her being 23 at the time.
The Joseph Yellowley who married Hannah Adams in 1781 in Woodhorn is probably not the father of the Thockrington William Yellowley, as William's brother John was born in 1787 with mother Barbara.
There was a Joseph Yellowley baptised on 8 Aug 1742 in Hebburn, son of Thomas. He was buried in Earsdon on 28 January 1831. Is this the one who married Barbara, or the one who married Hannah Adams?
There's also another Joseph Yellowley baptised 19 Feb 1749 in Hartburn, son of Thomas and Margery. He was buried on 13 Sept 1821 in Bedlington. Could he be the father of the Thockrington William?