So what is the plan now?
Plenty of choices – well a few.
Make a new nut.
Make a screw and nut.
Buy in some trapezoidal rod plus a nut (could make a nut).
Make an acetal insert for your present nut.
Probably others.
I would probably make a new square thread screw and nut if the screw is worn in the middle. If there is enough metal in the worn nut, I would use that – bore, bush and cut the new thread.
First thing is exactly how worn the existing nut/screw combo is. They will work perfectly adequately right up until they fail. Backlash is neither here nor there with a lathe – once the backlash is taken up. Remember, there will always be some backlash with cut threads (acetal, by the ‘heat and squeeze’ process will always be a very tight fit, needing some relief after moulding).