On LEU Treehugger, we used a 3/4" 4120 tube as the axle - heavier wall than the common 0.035, but not sure how thick. I remember reading of bent axles on LEUs prior to making the gear. We never had a bent axle, but did bend the main gear legs from hard landing impacts.
The wheels were high quality plastic spoked wheels, purchased from a bike shop. I believe we drilled the hubs out so the bushings would fit, but we may have selected bushings to fit the existing bore - just can't remember. The bushings were scintered bronze flanged bushings with 3/4" ID, OD may have been 1".
We drilled the end of the axle tube to install a cotter pin. Put washers on inner and outer side of the hub. We did not have brakes - brakes are for sissies. As long as you have plenty of clear space ahead, and grass helps.
May be able to make the washers correct thickness to eliminate side/side play, but Cessna4542c's method of threading end of the axle would work great, and install castle nut (maybe faced to make it thinner) w/ cotter pin.