I think you could learn it quick. As far as what to learn, I would say focus more on CAM/Gcode/CNC concepts vs CAD. When programming on the lathe you really just need a 2d view of the part wire frame so that it pretty dang easy. When programming on the mill you do usually need a 3d part model, but I would just hire that out. So in my humble opinion I think focusing on the CAM/Gcode concepts would be the best/essential route to transition in CNC manufacturing.
How big are your part runs? I don't run my own shop so I don't have spread sheet numbers to back this up, but I would think anything over 10 parts would be a good investment.
Machines:
I started on these prototrak machines. They aren't full on "CNC" machines because they don't accept Gcode programs, but you are still programming in a lot of automation.
ProtoTRAK SLX : ProtoTRAK CNC - Southwestern Industries, Inc.
ProtoTRAK SMX : ProtoTRAK CNC - Southwestern Industries, Inc.