The Opies, in "The Singing Game," discuss the rhyme group "Up Against the Wall," which includes "I'll Tell Mother" and others that commonly were put together to make a game. Some are based on once popular songs. A number of them are known on both sides of the Atlantic.

I haven't heard the medley, but the first and last mentioned by celtic blue5 suggest that Redpath, starting with a 'wall' verse and ending with a lad in 'americay,' may have been thinking of the migration of these rhymes across the ocean.
The Opies mention one of the "Up Againt the Wall" group collected in America in 1886:

"I like coffee and I like tea,
I like boys and the boys like me.
I'll tell my mother when I get home,
The boys won't let the girls alone.
O sweet beans and barley grows...."