I had to look around for pictures and found some. First, What a great looking little engine! I can't imagine any little 1950's - 1960's kid not being happy with a set like that.
With blind axle holes on the wheels it's entirely possible that these engines were designed to be thrown together
with zero thought to being repaired ( no need to measure the wheel spacing if you press them onto the axles till they bottom
out )
The back of the non geared wheel has a large diameter boss where the axle is pressed into ( apparently the same diameter as the hub in the front of the wheel )
I would design a tapered wedge type tool that fits on both sides of the axle shaft ( "U" shaped ) and only bears on the center boss on one side and the stamped sheet metal frame on the other. Hopefully you can rest the top of the inverted frame on a vise and gently tap the wedge between the frame and wheel center and press it off from behind. ( tap, rotate the wheel 45 degrees ,tap, rotate the wheel 45 degrees, tap, rotate the wheel 45 degrees, etc. )
I have zero knowledge if this will actually work since I don't own one ( you'll have to judge whether the sheet metal frame is strong enough to be pressed against ) but unless someone with actual working knowledge has a better idea, this is how I would try it.
Maybe provide some better photos of the bottom view between the frame and non-geared wheel, and tell us whether the frame is steel or aluminum?
