Having read newspaper reports on this incident I'm inclined to think it is a very sad story for all concerned.
William Dobson was born on 13th August 1898, the son of James Dobson, fisherman, of Greenway Street, Handbridge and Sarah Hannah Thornley.
James and Sarah married at St Mary's-on-the Hill, Chester on 12th August 1876. The parish register states that James' father is also named James and a fisherman, Sarah is illegitimate.
The 1911 census states that James and Sarah Ann (she is not using the name Hannah now ) have been married for 36 years and have had 9 children but only 7 are still living.
The newspaper reports state that William had two children. He was sentenced to death but as you said, there was a huge reaction to this by the people of Chester and the sentence was commuted to penal servitude for life. He served his sentence in Maidstone Prison but was released in 1936, after serving only 8 years.
Newspapers reports from 19th June 1936, state that on his release, his wife, children and 80 year old father met him in Liverpool.
'He is now in the Liverpool district with his wife and two children, where he has been found work. His son Jimmie, being only a few months old at the time of the crime. It his birthday this month and he is celebrating it with his father, who he does not remember.'
I agree with Philip and Garstonite that the marriage of Ethel Welch to William Dobson in 1922 is highly likely to be the right match.
Trying to find a match for children born to a Welch/Dobson marriage I think these are a match.
Alice Dobson born 1923 Chester - mothers last name Welsh
James Dobson born 1927 Chester " Welch
John J Dobson born 1925 Chester (died 1925) " Welsh
Sarah A Dobson born 1939 Chester (died 1939) " Welsh
James Dobson senior died in 1939. There is no mention in the newspaper report of his funeral of William or his wife attending but it does say his grandchildren, A Dobson and J Dobson attended.
James Dobson Senior made his last will and testament on 7th December 1936. He appointed his sons, James Dobson of 7 Greenway Street, Chester and William Dobson of 7 Greenway Street, Chester (presently residing in Birkenhead) to be his executors.
He left no. 9 Greenway Street to James and no. 7 Greenway Street to William.
I think there is a possible death for William in the Chester and Ellesmere Port district - quarter 1 - 1980.
A couple of interesting links concerning William Dobson below (I hope they work)
http://goo.gl/YWmLxKhttp://goo.gl/6lLrcRRegards
Trish