Here’s the information from the other thread as provided by Frank Jeal who’s sadly no longer with us to answer any further questions. I believe Tommy Sands may have got it from Brian Mullen who learned it from Frank.

“OK, Here's the story of "Mr Dunne" as I know it!    The song was originally collected as a pub song in Dorset by Richard Valentine , probably from the Dorchester area where he lived. Richard came up to Oxford in about 1959 . He was a fine folk singer and he and I occasionally sang duets at the Heritage club. Richard was loosely attached to the Oxford Uni. Morris Men , of which I was a member , and on a morris tour he wrote out the words of Mr Dunn on the back of an envelope. After the tour I went to learn the words , but found that the envelope had got damp and one line was illegible , so I rewrote that one line. That was significant, as to me that line is instantly recognisable , and acts like a "radioactive marker". I can immediately tell if the version in question has been 'through my hands'. I should say that all the versions I have seen in recent years including the ones on this thread, have my line in it ! When I went to Bangor in 1962 I took the song with me , and many of my friends in the Bangor morris may have learnt it. I then moved to Dublin where I also sang it around the pubs etc. One Irishman who had been in Bangor , and learnt the song there was Brian Mullen., from Derry. He admitted to me that he had passed it on to the Sands family, when I rang him , having heard their recording of it on Radio Eirinn.
So , the song was originally collected in England. I believe it was used in Dorset as a 'challenge' song, - if you could sing it you were still sober enough to be served !.
In recent years I was contacted by a genuine Mr Dunne who was writing a history of the (Irish) Dunne family, He had heard Cathal MacConnell sing it , and Cathal suggested he contact me for the words. So the words have now made it into a formal history of the Dunne family!!
What really matters though is that the song caught on, travelled widely and given lots of people a good laugh !!”