Even though there is not that much information on the inside of these Edison Bell sets, or even on 1920s crystal sets in general, some pictures could be found online that showed the inside wiring. These are of three different units, so that common features to these three are likely to be original factory fitted parts.
From these images, it becomes clear that the set was originally wired with 'thick' wire rod. This was actually quite common in 1920s radios and electrics; bare wire forming a neat three-dimensional circuit. From the pictures made a guess at the diameter (probably just over a mm) and got some tinned copper fuse wire of just over a mm diameter.
To straighten the thick wire ('rod' material, really); the regular procedure. Unwind and manually un-bend a length of the wire. Then clamp one end in a vise (or clamp to the bench-edge) and the other end in a drill. By pulling and then twisting the wire left and right it gets nicely straightened. A hand drill is good to use as it gives a bit of 'feel' for how you're deforming the wire and when to twist back. For safety, always keep a bit of tape over the wire ends (cut wire is rather sharp and springy...).
With needle-nose pliers and a bit of 4mm tubing as former, bent loops in the wire ends and made the sections to fit the terminal posts. The first leads to fit were the connections to the crystal detector - these must've been formed in-place, as they pass through the small hole in the base-plate and should have loops at both ends. The loops at the detector crystal-covers need some tweaking with a kink in the lead to clear the glass. (Suspect the original may have used slightly thinner gauge wire, or perhaps even not made a loop there at all - just a straight wire-end?)
Following the same procedure and consulting the reference images of original sets, the complete wiring could be recreated and fitted again. Also the leads to the slightly-different reproduction lamp-fitting was added.
Bringing the leads to the far end of the variometer seems a bit odd - reversing the variometer to have the terminals closer to the baseplate would've saved some copper. But perhaps the ease of fitting (or whim) was more important than a few inches of copper wire back in 1923.
With some thinner cotton-covered wire to the battery-compartment, the lamp again works too. (From the pictures it could be seen that this should also be thick 'wire' with a green insulating sleeve - but there's a limit to authenticity here, this is more practical.)