Here's the additional info I can provide:
The Punch and Judy Show by Gracie Fields
According to "The Complete Entertainment Discography" by Brian Rust:
Recorded February 14, 1930 in Hayes, Middlesex, England.
The Punch and Judy Show by Jack Payne
According to "British Dance Bands on Record 1911 to 1945" by Brian Rust & Sandy Forbes.
Recorded February 8, 1930 in London, England.
The Punch and Judy Show by Layton and Johnstone
Recorded approximately end April 1930.
But the song was copyrighted in 1929 (Chappell and Co) and most likely first published on sheetmusic.
The earliest recording I found in "British Dance Bands on Record 1911 to 1945" by Brian Rust & Sandy Forbes.
Harry Hudson's Melody Men
Recorded December 24, 1929 in London.
Released on Edison Bell Radio 1317.
After this, in January and February 1930, various versions were recorded by the likes of Harry bidgood, Ambrose, Stan Greening, Charles "Nat" Star, Bert & John Firman and Jach Hylton.