I've only ever heard the one verse sung to the Red Wing tune, but the wording varies in places. See:

CHARLIE CHAPLIN -The Digital Tradition file. Taken from an Oldham Tinkers record. Only the first verse belongs to the Red Wing tune, the rest of it having been cobbled together from various children's playground rhymes that happen to mention Chaplin, and which are usually sung to the first part of In and Out the Windows.

Lyr Req: Redwing - East Anglian Song and Tune -Three slight variants and some discussion.