Remember it won't be a flat circle because bike wheels are wider at the hub than where the spokes meet the rim, so it will actually be a very shallow cone (sort of yarmulke shaped :) ) To make one of these out of cardboard you'll need to cut a small slice out (like Pacman with his mouth nearly closed) and join the cut edges together.

I betcha what would be even beter would be to make them out of coroplast!! more weather-proof than cardboard, and a little bit lighter!

Looks like the circle(s) were made, painted, and then placed in each side of the wheel and tied (sewed ? ) on with some sort of twine when used the spokes to hold both halves on.

From the off center looks of the center hole it wasn't made perfectly. The outer circle would be most important anyway. Can you ride one of these on a windy day? Corroplast would be the way to go. Look for old corroplast election posters.

Like a carpenter, just nail or pin the end of the string at the center of your circle. Tie a pencil at the free end of the string at the length you need it and go around and trace the circle holding the string taut. Do the same thing with the smaller diameter hole. Cut on the line with a razor knife. Use scissors for final trim when on the wheel. Trace a line for where you are going to "sew" the covers. Use an awl to poke the holes where you will sew the covers on. Decorate, coat with varnish or something and voila.

To make the circle template take half the rim diameter of string and pin to one end on the board then attach to a pen, draw a perfect circle the right size in seconds... Same trick for figuring out where to go next in your plane... Maybe make this one plastic board, or for the crazy factor print it on a big transparency and laminate for extra life, make a crazy design that you can see through instead of white bits or brown in this case.

Looks like it's held on with zip-ties. From the pic, it looks like an art-piece, not a real-world item-you could try coroplast(?), a corrugated plastic cardboard (but you would probably have to slit it so that it would take the cone shape of the spokes) so that it would survive wet/damp weather. Easiest way to make the circle would be to trace the inside of the wheel rim-you have to take it off anyway, why not use it as your template?

for a perfect circle
this

you probably wouldn't need wood, you probably could use cardboard as the rectangle