I don't know the particular lathes, I'm afraid but can offer this:
The half-nut control is very unlikely to have relied on friction to hold its selected position.
If the half-nut control relied on a sprung detent that has absconded, there ought be traces such as hemi-spherical drillings, probably somewhere on the apron, to accept the sprung ball or plunger.
I have just a quick shufti in the Fount of Knowledge – Tony Griffiths' 'Lathes.co' web-site – and unless another contributor can answer directly, I suggest seeing if the Perfecto "chapter" therein gives any useful details. Its photos of restored Perfecto lathes clearly show the clasp-nut lever sprouting from a large disc, and I suspect as you do, that this disc holds a detent working in aforementioned drillings. Or would, if not missing from the lathe receiving attention.
Examine the part of the outer face of the apron normally covered by the disc, and see if it has any features that match ones on the inner face of the disc. The missing bits might be as simple as a short coil-spring and a ball or short pin, housed in a blind hole in either the disc or apron, and mating with said hollows in the other.
I did not search further, but Mr. Griffiths may have facsimiles of the Perfecto lathe manual for sale. I have bought corresponding ones for my machine-tools, from him.