"For the record, "Singing the Fishing was aird on BBC in 1959. Before the end of 1960, the song was well known (and badly overdone) in Greenwich Village folk circles".

Not according to the BBC, Dick:


"Singing The Fishing
First transmitted on16 August 1960 .... Singing the Fishing won the Prix d'Italia for radio documentary in October 1960 and was eventually transmitted in 86 countries".

http://www.bbc.co.uk/radio2/radioballads/original/singingthefishing.shtml

I find it unlikely that the song should have been "well known (and badly overdone) in Greenwich Village folk circles" weeks after its first broadcast in the UK, and even more unlikely that Llewyn Davis could have made a recording of it when he was 8 years old! The annoying thing is that there are innumerable traditional fishing/maritime songs the Coen brothers could have chosen to be Llewin's father's favourite song, which was supposed to have such an emotional effect on him in his care home (though why they had to have his father shit himself is another question)! After all, if they could (allegedly) base Jim's beard on that of Paul Clayton, there are plenty of Clayton's maritime songs that would have fitted the bill!