With all lathe leadscrews, there is backlash. A small amount of backlash is even necessary for humans to know and trust the move they make has some certainty. For CNC, we are now talking servo feedback control, for which backlash is untenable. If a servo motor drive starts moving, even by microns, and the feedback is still not reporting movement, the drive energy racks up rapidly into an instability oscillation that can rattle your teeth!
Step #1 for any any conversion to CNC is to replace the leadscrews and nuts for a ballscrew drive which has zero, or near zero backlash.
Adding CNC to a ready-made lathe infrastructure gets you end up with a sort of CNC machine. You will have invested in the CNC hardware, a real amount of adaption design and learning, and the time and effort to install and calibrate. For many at HM, much of this would be very cool fun to do, and succeed at. The added value to you is that in getting to that point, you also end up with a deep understanding of CNC and your machine. This state is different to one who can operate a CNC machine by training.
I am not sure what value one would put on undertaking and completing a conversion + conversion hardware costs. The raw cost of a purpose-built CNC machine is likely to exceed $4000, even a used one, and you would not know exactly how it works, and would still require the learning curve to use it.
CNC adherents in this community do all kinds of conversions and also "from-scratch" CNC builds. Some will purchase linear bearing precision slides, and ballscrew drives, and make their own stable beds from steel reinforced epoxy-granite in home-constructed plywood molds. The machines are often individual, unique things of beauty. A regular machine, without CNC, is designed around the fundamental philosophy that is part of a scheme that includes an advantageous, built-in, twelve billion cell human brain (+ other soft parts liable to damage). True CNC, taken to the point it's strengths are properly exploited, involve higher powers, speeds, and motions a human cannot do, + they never stop to bring out the micrometer and make a measurement.
It's interesting, and looks like fun, and clearly we have a load of experts here, but honestly, somewhat out of my league right now. I do aspire to motorize the cross-feed on one of my lathes, and I already know I can use a servo drive main carriage leadscrew instead of using a set of change gears. Not true CNC, but easy enough to do.