Will be great to see that lathe back as it should be!
Helical back-gears… Was that a Pittler speciality? That’s the first time I’ve seen that on a lathe.
Advice, you ask…..
Another source of information is Tony Griffith’s lathes.co.uk site
I’d keep the two lots of metal as you’ve photographed them, separate from each other for the moment.
Strip, clean, repaint, re-assemble the lathe as it is, also clean all those loose items as a separate lot.
That will then make it easier and cleaner to decide in that heap of bits, what is a lathe part, what is an accessory (including change-wheels) and what is neither (or you can’t immediately identify).
Examining the heap, I can see:
A type of chuck or tap-holder – at top right, next to a smaller item that might be a die-holder.
In front of those, one journal and bracket for a countershaft I guess originally used with the lathe, and what seems to be a hand-turning tool rest that might fit what may be a socket visible on the front of the saddle.
Next to the tool-rest is a handle that is probably for the cross-slide – it looks a brother of the top-slide handle still in-situ.
Beyond that I see three chuck-jaws, two Stauffer grease-cups, a small lever that could be for the half-nuts mechanism or something similar; and at the back, what might be a bearing half-brass.
I can’t identify the two big castings in the left foreground, the bracket with the worm, and the item with what looks like a plastic pulley (a milling-spindle?).
I assume the shaft with the universal joints is part of the lathe’s power-feed but it really needs an expert on Pittler lathes to identify it correctly.
At least some of those gears may be change-wheels, though using helical gears for that seems unusual. The small pinion in the middle of the heap of gears might be the worm-wheel for that (power-feed?) shaft.
Many of the small items might not be lathe parts, but this should become clearer once you have cleaned everything and started to analyse what fits where… or does not fit anything.