Paddy Backwards is probably the most common title for this nonsense song, a very common genre of song in previous centuries.

All of these titles are used and they should be considered all one song.
Adam Ham Town
Crazy Song
Paddy's Ramble to London
Nottamun Town
I'm going up to London
Noddingham Town
The Lying Song
Fair Notamon
Fair Notamun Town
Fair Nottiman Town
Notamun Town
Nottaman Town
Nottingham Fair
Old Gray mare

Found on both sides of the Atlantic all of these versions seem to derive from the Pitts broadside 'Paddy's Ramble to London' of which unfortunately I haven't got a copy, but the first line is 'Said Paddy, In Ireland no longer I'd stay'. If anyone has a copy of the broadside text I'd certainly like to see it. Roud number BTW is 1044.