A fellow forum member suggested putting the gear centre hole to a drill of the same size, lower it to the vise, and clamp it down. Once the spindle is centered to the hole, swap in the bigger drill to drill through.
I did the same before on a project but used steel dowel pin instead for better accuracy. Doing the same thing on the lathe is very much easier if the run out is within satisfaction.
The young gentleman dropped me a pm and we made arrangement to meet him at my shop this morning.
The job was an easy one but I was still careful in drilling with incremental size drill to 8mm for the first piece. The 4-jaw self-centering chuck was used. I really love the 3 and the 4-jaw chucks from Rohm. Their run outs are well within what I normally require. With the first piece done, the 2nd one was completed much quicker. Both fit the wheels well, which is where they are meant to be..
The axle was next. For this 4mm steel pin, there is a short section of 3mm diamter on each end. He wants each of the short sections extended to 7mm for the wheel. Not a difficult task as well.
Both were done rather quickly and tested. I've forgotten to take pics during the operation as we were chatting away. This pic was the only one I took when he was about to leave: