Not an answer, just some clues, perhaps, to the use, "to be noncommittal or inconstant".
Compare: wiffle-waffle, whiffle-waffle
The first (English Dialect Dictionary, Joseph Wright) gives a definition of wiffle-waffle "to whet a scythe", which I understand (perhaps incorrectly?) to involve a back-and-forth motion. Wright finds the attestation on page 141 of The Dialect And Folk-Lore Of Northamptonshire by Thomas Sternberg. London. 1851
Wiffle (whiffle) and waffle (whaffle, woffle) seem to involve swinging or swaying or moving back and forth (literal 'vacillation'); the word wiffle-waffle has among its meanings 'to speak in a meandering manner'. In the American use of 'waffling politicians', they either speak in an evasive manner so as not to be held to an opinion, or they flip-flop and say today the opposite of what they said last week.
The second instance from Dizionario Delle Lingue Italiana Ed Inglese, Giuseppe Marco Antonio Baretti, (1839), gives a definition for whiffle : to swing, to sway back and forth (muoversi ondeggiando...dondolarsi). [But this link was mere lagniappe :-) ]
See also whiffle-whaffle : "a person of unsteady, vacillating character" in
The Dialect of Craven in the West Riding of the County of York (1828).
P.S. In his A Glossary of the Cleveland Dialect (London, 1868) the Reverend J.C. Atkinson defines waffle as "to waver or vacillate; to be undecided" and refers to Old Norse vöflur (doubt, uncertainty, hesitation).
(more lagniappe) Here we have a description of whiffling-whaffling that shows it was a skill akin to modern American baton-twirling at high-school and university football games.
Here in Fenland Notes and Queries (Cambridgeshire, 1900) we have an attestation of waffling as a kind of deceitfulness ("tricksy, waffling fellow").
And I think we have a 1955 Canadian attestation in the House of Commons Debates, Official Report (Volume 5) where the "slang expression" waffling is explained as follows:
There has been a great deal of "waffling", to use a slang expression. There has been a great deal of indecision and a great deal of uncertainty. No one knows just where he stands.
There looks to be a British-English attestation of the meaning "deliberate evasiveness so as to avoid being held to an opinion" in the record of Parliamentary Debates in the House of Commons for 1953, but I cannot see more than a piece of the paragraph:
...and myself in the art of waffling, but at the present moment I am asking him a direct question and he is not giving me a straight answer. [my emphasis]
And in Medical students and medical sciences: some problems of education in Britain and the United States (Oxford University Press, 1955) we find on page 67:
This last is an important aptitude, and in Britain has received the name of 'waffling'. There are many techniques — the red herring, the evasion, the ambiguity, and so on [my emphasis]
N.B. These last two attestations go to your question about "divergence".