Use of 'what?' to end an assertion

Solution 1:

I've no reason to think this usage was ever particularly "colonial English". I suspect it probably originated among Victorian gentry in the motherland.

Moving on to total speculation, I think it most likely started as a shortening of "What do you say?", "What do you think of that, then?", etc. Effectively, it's no different to modern usages whereby we append an interjection such as "Huh?", "Yeah?", or (really modern! :) "Innit?" to a statement.

I also suspect it was never particularly common at any time or place, except as a facetious usage.

Solution 2:

The OED has this as sense 4.c. of "what, pron., adj.¹, and adv., int., conj., and n.": "...an interrogative expletive (sometimes with eh)... esp. in recent trivial or affected colloq. use".

First attested use? A diary recounting the Mad King's ravings:

19 Dec. 1785 F. Burney's Diary, Vol. II, p. 398:

[George III] said,—‘What? what?’—meaning, what say you?.. ‘..it is not possible. Do you think it is?—what?’