I don't think I'd ever use wangle (not wankle!) to describe moving objects around - it's always used of human interactions as far as I'm aware. And I've used winkle as a verb as well so it's not just US English, and assumed it might have something to do with the edible winkle (the shellfish - mollusc?) which needs a pin to tease the edible bit out of the shell.
Wankle on the other hand sounds as if it should exist as a word but I've never heard it used. Wanker, yes - but let's not digress.