Whence these terms?

I can trace Pooh-Bah back to the character in Gilbert & Sullivan's Mikado (c. 1885), and Pooh Bear back to Winnie the Pooh (c. 1920). But I doubt that the latter came from the former, as their personalities are entirely different. My suspicion is that the share a common, prior source. Does anyone know?

Related, and possible clue: I recall reading that many of the character names in The Mikado were taken from the baby-talk used in British nurseries: e.g., the character Pitti-sing is baby-talk for "pretty thing". I have no confirmation for this, however.