…lathe in threadcutting mode…
A modern take on the threadcutting aspect, especially if it would be difficult to set the lathe up for the coarse pitch the object requires, would be to 3D print a facsimile of the object and use it as a copy template (like how rifle stocks were duplicated from a hand-made master pattern).
The helix cutting above reminds me of a woodworking contraption, I think called a router lathe, that used a tensioned steel cable over different size pulleys to synchronise rotary and sliding movements. I guess in that situation, some potential slip would not be a disaster, unlike in threadcutting.