The nut only has to be a "close fit", it most likely will not screw on due to different threads and might be better to find a nut that is slightly smaller diameter hole than the shaft and then drill the nut to fit tightly on the screw (slide over screw shaft) then weld in place.
Too large of a nut opening and it will be difficult to center the nut on the shaft for welding and you will end up with a wobbly out of center nut.
Once welded, get some grease or oil on the screw shaft, dry shaft is most likely why the nubs worn out.. Takes more effort to turn the screw without some lubrication and that extra effort wears out the nubs..
