Wrong song, Tim!

The Erin go Bragh in question is a Scottish song which appears on broadsides of the mid 19th century, and was evidently quite popular; it likely dates from that time, though it is usually sung to older tunes. The most usual location (and presumably the original) is "Auld Reekie", but there are also copies with "Whitechapel Street" and with the hero's name changed from Duncan Campbell to Pat Murphy, which rather spoils the irony, not to mention making the song a little nonsensical in places.

See  Bodleian Library Broadsides:   Erin go bragh

and  Glasgow Broadside Ballads:   Duncan Campbell

Once you've lost the Scottish location, the song loses much of its point, really. Perhaps for that reason, it doesn't seem to have been taken up in English tradition, and has been found mostly in Scotland and Canada (occasionally in the US; Cecil Sharp found it in the Appalachians as "My name it is Clay Morgan from the town of Hogyle").

The Roud Folk Song Index lists a fair few examples at number 1627 (Laws classified it as his code Q20) with only a couple of Irish examples. One (Armagh) begins "In London one day as I walked down the street;" another (Clare) "My name is old Paddy from the town of Athy". I haven't seen those, nor the De Marsan (New York) broadside, Daniel Campbell's Trip to England referred to by W Roy MacKenzie (Ballads and Songs of Nova Scotia, 330-1), so I don't know how they treat the story.

This song should not be confused with other, unrelated songs that happen to include the well-known phrase Erin go Bragh.