I found myself in a similar position a year or two ago. A neighbour gave me an old saw, which I thought I could put back into usable condition. It was rusty and a previous owner had got as far as removing the split nuts, which of course were not with the saw when I got it. Fortunately in this case the screws were not about to shift from their long-held positions.
In what looks now like an embarrassingly crude job, I found some suitable bits of non-ferrous metal (obsolete foreign coins from holiday change!) which I drilled, tapped and filed down. I think the tap was probably 1/8" Whitworth (as that's what I had) and it was close enough to fit, in a strictly utilitarian, not-proper-engineering sort of way.
I also filed the teeth so it cuts again.
I did it at the time, not to get a pretty saw, but to practice on and learn what I could do. However, the nuts do work - which is the main point really - so the handle is comfortable and stable and the saw is usable.
The maker's name is just about visible in this blurry photo as "Russell and Horsfield" of Sheffield.
I've since learned (from Simon Barley's book) that their partnership was shortlived, from 1854 to 1856, so (assuming that they did not carry on using the old name) this saw is probably about 160 years old. That makes me feel pleased to have saved it from the bin.
Carl, I think you could do a similar, or better job on your saw!