Just listened to the new Chris Foster tape Sting In The Tail.
There are two songs on it excellent for younger kids: Herring's Head, listed as traditional, and The Neighbour's Cat, written by L. Rosselson, whoever that might be.
As to the Alice poems set to music, I don't know if they came from Disney but I suspect my mother knew them before the movie came out. I can still sing "You Are Old, Father William", but have forgotten "Speak Roughly To Your Little Boy" and "' 'Tis the Voice of the Lobster".
I'll ask her.