I cannot thank you enough for your input ... I am certainly starting to think outside the square! I did start to try and pick out some likely suspects re the death of various Joseph Ashtons, but was forced to abandon the search as being too broad especially as I know very little about where he was born and his date of birth etc.
Based on your earlier suggestion, I am now starting to wonder, if in fact, William Wilcock b. ~ 1856 Bradford YKS might be the son of Joseph Ashton and either Ann or her sister, Unknown Frankland. I realise that Ann was already married to Joseph Wilcock at the time of 'my' Thomas William's birth, but an affair could've occurred between Ann & Joseph and the result was William - who was reared by the Wilcock family. Similarly, Joseph could've had an affair with Ann's sister and for some reason, Ann & her husband agreed to rear William. And as suggested earlier, perhaps William did not take on the name of his father until he was joining the Merchant Navy. Certainly the name of Frankland is within the family. As my sister pointed out to me Thomas William's eldest son, Joseph William, named one of his sons, Frankland who was known as Frank.
By the 1881 Census, William Wilcock b. Bradford, son of Joseph Wilcock seems to have disappeared. There are two William Wilcocks listed b. Bradford but they are sons of a John Wilcock in both instances.
Thanks also Z for your interest in Horton House and locating it on the census. My sister has also been busy trying to find out about Horton House and has come up with the following:
The draft conservation assessment for Little Horton Lane has a map showing the location of Horton House and another publication, a book called "Rambles Round Horton" by William Cudworth, published in 1886 contains a wonderful drawing of the House and also contains the following references:
p.93" Horton House remained in the hands of the Fitzgeralds until a short time ago when it was disposed of by public auction." There is no mention of Joseph Ashton in the book.
p. 226 ".....the 'Ashton Dole' is a charity, the proceeds of which are derived from property left under the will of John Ashton in 1712 to be distributed 1/2 yearly among poor people of Horton above 60 years of age who are not in the receipt of Parish relief."
Back to the drawing board ... again many thanks