I first heard "Brian O'Lynn" on Wally Whyton's Folk Music Broadcast in September 1984 along with I Wish There Were No Prisons, both being sung by Jim Elvin (my spelling). He had a delightful style in accompanying himself on the fiddle. I transcribed it for use by the Idle Fellows and we played it fairly often. This version is entirely English and is quite different from the Irish versions I've come across. Judging by his accent, Jim Elvin was probably a Yorkshireman, the give away being his "'Ey up or I'll stick ya!".
Since writing the above in the late 1990's, I have now (April 2013) come back to this song in the course of my studies of Tyneside folk music. Northern Garlands give us a song called Tommy Linn published The North Country Chorister (1792) with 9 rollicking verses very much in the same vein as the version I sing. I have also determined that the name "Jim Elvin" was indeed a misspelling the gentleman's real name is Jim Eldon. Pace Jim!
In July 2013, I came across Ireland Sings by Dominic Behan in which he made the following remark:
One of the best known nursery rhymes. Tom O'Linn, Geordie O'Linn, Jack O'Linn, agaus alainn againnse, and that's where it might have derived from, 'Ol' drink. The super optimist - badly needed just now.
Behan's version has 4 very Irish verses about courting and marrying a daughter and mother all sharing a bed and falling in water when crossing a bridge.
Further delving in November 2014 has turned up an alternative title namely Lockington Fair which is given in the song and ceased generations ago.