The roll of wire looked like a Slinky, but was not a Slinky.
It was wire used in some form of gardneering.
After lots of measuring, bending, re-bending, deciding some kind of plan might be a good idea, I eventually ended up with this:
Obviously it's a little further on from the bend, unbending stage.
The key to this whole machine is the guide plate across the centre of the housing.
This keeps the push-rods where they should be.
Here's the test of the mechanism:
I had intended putting something 'fluffy' on the ends of the push-rods, but all I could find were skulls:
The finished item now looks and moves like this: