I would guess some kind of coil or armature winder - I saw coils for electric motors being would by hand in Hong Kong in the 1980's..
The swash plate at the handle end moves the chuck arrangement forwards and back each revolution, so spreading the wrapping over about an inch... My guess that the missing piece is the bobbin holder, and it is operated by the cam device to aid the 'bale' arm that revolves as if the object being wound is not circular, the feed speed and thus the tension in the wire would vary... Maybe with a fixed bobbin the wire broke, so it was made to move as the wire was being unwound... I say wire, it could equally be a thread binding of some sort.
The shape of the chuck could be a clue - the object has to fit into the two arms - question: does it also need to project beyond them, or even pass over them???? Try fitting something like a short length of toilet roll tube onto a wooden mandrel to form a cross with arms projecting out of the forks, and putting a thread through the bale arm - it seems to me that it will wrap some sort of figure 8 at right angles to the axis of rotation...
The gearing appears to be about 2:1 - the bale arm revolves once each side of the chuck - as it moves about 1/2" along its axis - this makes me think the object being wrapped must pass over the arms, or the arms would also become wrapped.... It isn't something as simple as a string winder??
