The motor that you picture on the Clarke milling attachment to the C1 lathe looks identical to the motor on my X1 mill from ArcEurotrade.
Since it is unlikely that you would need to use the milling spindle and the lathe spindle at the same time I suspect that the attachment does not have a separate power controller board and that it is run from the same controller as the lathe motor. Is there a socket on the back of the C1 that the attachment plugs into?
I would urge you to consider making a belt drive conversion rather than replacing the gears in the mill headstock. The belt drive is much quieter, smoother and reduces any shock loading on the spindle. It is sudden shock loading that tend to upset the controller board. My belt drive conversion on the X1 mill is shown here:
http://mikesworkshop.weebly.com/mill-belt-drive-conversion.html
As an alternative motor you may be able to pick up a small dc motor that was made for an electric golf trolley. Ebay usually lists some of them. This could be used with a 12 V motor controller from Ebay
Mike.