Welcome to PF.
You can do the entire job in almost any metal lathe. Ask around to find someone local who has a lathe.
Preferably a hobbyist, maybe a school, garage or ask a machine shop who has one in your area.
1. Maybe start with a solid steel 5/8” or greater diameter extension shaft. The larger diameter will provide more wall thickness for clamping to the motor shaft, it will also stop the wheel sliding one way along the extension.
2. Bore a 5/16” hole (=7.9375 mm) in one end to accept the motor shaft. Hold the job in the chuck, with the drill in the tailstock. Drilled holes are often oversize, hence the 62.5 micron allowance. Check the fit with the motor shaft, polish out the bore to enlarge it if needed by using abrasive paper on a rod or wooden dowel. Drill a couple of holes and thread them to take a copuple of clamping screws to the motor shaft.
Note: If you cannot get a good fit to the shaft you might be better using some form of flexible coupling at the motor shaft junction.
3. Reverse the extension shaft in the lathe chuck. Turn down the shaft to just over 1/2” diameter over the section that will carry the wheel. Keep the fit tight over the part where the wheel will seat. The wheel's plastic hub will not handle a huge internal pressure so the key is needed to stop rotation. Make a step to prevent the wheel sliding along towards the motor.
4. Machine a step at the far end of the extension shaft to be a snug fit inside the free end support bearing.
5. Cut a square key groove in the shaft for the wheel hub key by running a tool steel cutter backwards and forwards in the lathe saddle, with the chuck clamped to prevent rotation. Advance the cross slide very slowly as you cut the key groove progressively deeper. File a rectangular key to fit in the wheel and shaft keyways.
6.You will need to find a way to stop the wheel sliding, off the key, towards the bearing, maybe make a sleeve to fit between the bearing inner race and the wheel.
Good luck.