These display replicas needed to be made on such a short timeline, that the techniques employed required hacking.
Had there been more time, designs could have been drawn up, things could have been fabricated exactly to scale, etc. However, the timeline dictated a more advanced starting point and strategy: find things that are close in shape and size, and cut and glue them together with minimal fabrication of major components.
Ultimately, Explorer 1 was made from a 6 inch (nominal ID) PVC pipe, some plastic doll forms, a Christmas ornament, two air duct adapters, a cake ring, screws, stainless steel rod, fiberglass, and plenty of epoxy, glue, and paint.
Sputnik PS-1 was made from two Plexiglas (PMMA) security dome mirror (near-)hemispheres, aluminum rod, aluminum tubing, aluminum extrusion, aluminum strips, plastic automotive molding, PVC pipe, fiberglass, epoxy resin, screws, and chrome paint.
Regarding the chrome paint, I've tried a lot of them over the years with varying results. There was no time to do a lot of testing for this project, but fortunately I found someone who had done the testing already and published it. I credit this video for the final choice of chrome paint (and related surface preparations).