Hello and many thanks to all who have collaborated in this research; without you this mystery would never have been solved!
Do you remember there was only 2 William Douthwaite(s) left on my list?
They were: the Farmer (#10) and the Pipe fitter (#8)...
Rosie found out that the Pipe fitter from Redcar (Yorkshire) had emigrated to the United States of America in 1913 with his wife Hannah Mary Robinson, and had died there in 1940. (Congratulations Rosie!)
We were left with the Farmer; but he was an improbable candidate because he was married. Yet we could not place his death.
Well, it turns out the mysterious William Douthwaite was one of the two farmers born in Westmorland in 1871!What I have found out this morning from one of my relatives is that he was a farmer, who did work at sometime in Liverpool where he purchased several properties, and that he did marry previously to 1947, but this marriage was never consummated (although he did support his first wife financially over the years).
A happy ending for a difficult research...

Now I wonder... who was his mysterious first wife?

Once again, thank you all!!!