At the moment I have just one 4-wheel horsebox deserving of white wheel rims. For the future, I can imagine seven more axles needing the same treatment. So it is not worthwhile me building a special jig.

I have supported the model in soft padding, this is a clean face flannel inside a sandwich box. Then fitted a soft rubber mandrel into the mini-drill.
In operation I am holding the mini-drill in one hand and letting the mandrel spin the wheel set while I hold the brush in the other hand. My mini-drill is a Dremel 'Stylus' and I am using this on its lowest speed setting.

This is one of my results.
Some white paint has gone onto the ring inside the tyre and I can try to touch this up later. I think the concept is right; using a motor to spin the wheelset does give a better result compared to turning the wheelset by hand.
For a loco I would try supporting the chassis in blocks of foam or even a vice, and driving its motor through an unmbilical. I think this would give better access for the brush compared to a rolling road.