The Unfortunate Rake seems to have originated in Dublin around the 1790s; it later became popular in England and Scotland, and subsequently spread to the New World. The song was frequently adapted to suit different places and circumstances, and sung to a number of tunes; versions include The Unfortunate Lad (or Lass), The Young Sailor/Girl Cut Down in His/Her Prime, and many others. A Lyric Search of the DT for #350 will get you 26 examples. Of particular interest would be:
THE BAD GIRL'S LAMENT -a version from Canada, which names St James Hospital.
ST. JAMES HOSPITAL (Texas)
THE UNFORTUNATE RAKE - a 19th century broadside text which again names St. James Hospital.
The Traditional Ballad Index has some useful references, too; try St James Infirmary and Bad Girl's Lament
None of this would suggest that St James Infirmary does not belong to New Orleans, but it would certainly be a waste of time looking there for a hospital of that name!
The "Celtic" feel of the version mentioned earlier would be a result of the arrangement, I should imagine; though available evidence does point to an Irish origin, the family of songs that sprang from it rapidly moved beyond so narrow a classification. Incidentally, I'm not at all convinced that there's any relation between the tune of Streets of Laredo and Rakes of Mallow; anybody care to comment further on that?