Azizi, "Deeds not words" quotation from Fletcher need have no bearing on the rhyme. It seems to me that the thought behind that could easily be 'thunk up' by different writers. On the other hand- there's always the possibility that the writer of the rhyme in Halliwell knew of Fletcher's work. That is an 'indeterminable.'
Halliwell was a pretty good researcher; he leaves it without an author, and 'author unknown' is better than unverifiable speculation.

More interesting to me is the variation found in Talley. Halliwell's "Nursery Rhymes of England" went into several editions and was popular in America; the variation shows that his rhymes reached to African-Americans- possibly in slavery times.