Cut tire material oversize. Add an extra inch of length for each 10 inches of wheel diameter. For instance, if you have a 10-inch wheel, the appropriate length of tire material will be enough to go around and overlap 1 inch. If re-tiring a 50-inch wheel on an ordinary bicycle, the overlap will be 5 inches. Ensure the cut ends are smooth and square. Use a PVC tubing cutter and practice. For small diameter wheels, you might try to put a very slight bevel in the cut so that the length of tire along the wheel is very slightly less than the length along the outer diameter.
If nothing else, my procedure sheds some light on how the Racycle Crank can turn the seemingly simple into an involved exercise. It might get simpler with practice.