The 1937 Noel Gay musical Me And My Girl spawned three popular songs: "The Sun Has Got His Hat On," "Leaning On A Lamp-Post" and "The Lambeth Walk." The title of this last comes from a street in Lambeth, South London. With lyrics by Douglas Furber and L. Arthur Rose, the song is the finale to Act 1. It is performed by Bill, Sally and Ensemble.
The song contrasts Mayfair - an extremely expensive area of London even in the 1930s - with the working class Lambeth, though it remains to be seen if the skies are any bluer or the grass greener in Mayfair.
The phrase "Not like you but a bit more gay" should not be misinterpreted. At the time this musical was written, the word gay meant happy, lighthearted, etc., and would have had no sexual connotations for the audience.
This is a song that lends itself readily to parody, and has been used in at least one advertisement, for a travel company.
Suggestion credit: Alexander Baron - London, England, for above 3