They didn't remove or refit the wheels, as I took the new S/S wheels to them to have a whole new set of tyres. However C&S know their vintage cars, historic racers with knock-ons etc. Balancing is done with a set of cones, which centre the wheel on the normal balancing machine.
I do have the MWS tool for removing the octagonal nuts, which is miles better than the standard "spanner", so if I was going to a different tyre shop I'd take it with me and show them what to do. They don't really need torquing up afterwards, as they are self-tightening.