I agree with most of what Chris said above, although I'm still unclear whether you need a single slow linear traverse of the core or multiples as I described above. But the rest of what he says is pretty much right on.
What is clear is that while a mechanical winder can be fairly easily made to wind the simple basketweave/honeycomb coils shown in the videos, the need for an adjustable, synchronized linear movement makes a mechanical winder get very complicate very quickly. Six inches of linear movement is too much to be practical for a cam; a lead screw arrangement would be more suited, but it would have to be reversible and be geared to (and in a ratio synchronization with) the core rotation.
I do have the capability to cut cams with any motion contour, linear or otherwise, and a conventional winder was going to be a fun project for me, but I'm now thinking that I need the versatility of a dual stepper motor winder. One motor to drive a lead screw for both the small and large cyclic movements, and the other to rotate the core. All of a sudden this little side project is going to be a major distraction.
If I had the original in hand it would be easier to understand, but I don't own one of these - the photo above is not mine. I want to replicate it not just functionally, but so that it looks original as well. I was hoping that someone would recognize this wind as something used in vintage chokes, but it seems like an oddity. BTW, this theremin was circa 1960s-70s.