The tune MMario found is a reel, more usually known as Paddy on the Railroad (in America) or The Merry Blacksmith (over here).  "Paddy on the Railway" as given in the DT is a quite different thing.  I remember the song Fergie mentions as "Working on the Railway", or "Paddy Works on the Railway", but that wasn't sung to the reel, of course.  There's another song, too, "Paddy on the Railway"; there are several broadside copies at the  Bodleian Library.  Here is one:

Paddy on the railway  printed between 1858 and 1885 by W.S. Fortey, Monmouth Court, Bloomsbury, London.  That one doesn't fit the reel, either, and looks as if it should have a 3/4 rhythm.

Malcolm