Karen,
You might find it useful to look at the versions of Roud 2 on the Full English search on the Vaughan Williams Memorial Library site at EFDSS, particularly the 4 or so entries that show the info on 'The Unfortunate Rake' and how it somehow got attached to 'The Unfortunate Lad' Kidson noted down the version from Kate Thompson in 1892. At one point 'Rake' tune from Kerr's Merry Melodies is displayed above Kate Thompson's tune. I can't read music so can't comment on any similarities. I can't find any other British versions pre 1900 that mention St James's. As others have stated just as there are lots of Lock Hospitals there are lots of St James's (One here in Yorkshire at Leeds) It's futile trying to pinpoint which one it might refer to.

Was St James by any chance the patron saint of STDs? Baring-Gould would have known.

BTW there is a particularly graphic/gory Sc. version in Greig-Duncan.