How about this:
Get a section of old flat conveyor belting. Cut out as many sections as you need to make up the thickness you want. Cut the ones for the "groove" sections to a diameter suitable for the bottom of the groove, and cut the rest to your main diameter. Then cut out the centers to your rim diameter.
Now here comes the fun part: cut them all in half (or if the belting wasn't wide enough, start with twice as many semicircles). Stack them on your rim, staggering the joints (90 or 60 degrees, whatever makes you happy). Then fasten together. Possible with a few carriage bolts or sections of threaded rod through the whole stack, maybe some construction adhesive or something like that. You might even be able to simply nail them.
And you'd be able to make up a smaller diameter than the original tires if that's what you want.
As an alternative, I saw a nice scale "army jeep" at a tractor fair someone had made up using an old mower chassis, he even kept the mower deck!