I do feel like it's more than should be there, definitely more than a few thousandths. I assume the lead screw is hardened steel and the little nut at the end is brass for the cross slide... Wouldn't the brass nut wear out and not the lead screw as brass is much softer? I'll try to get those measurements soon when I have the time to do it.
Hard stuff up against soft stuff? So which one takes all the wear? The answer might surprise you!
Hard stuff will cut into soft stuff given enough pressure, which is achieved by making the pressure point sharp. This is as true for HSS steel cutting into brass as it is for tungsten carbide getting an entry into HSS.
BUT..
The mechanisms for wear and abrasion are completely different. Consider your (non-electric) car crankshaft. It's bearing surfaces are made really hard, and have to be ground to their finish. Then consider the bearing metals. They are made of white metal, a mix of tin and alloys copper, antimony cadmium, zinc or silver. They are so soft you can mark them with a fingernail! So long as there is a film of oil, wear is minimal.
Sintered bronze, brass, and other alloys are deliberately chosen to be up against steel. If the bearing journal is cast iron, there is enough graphite in the metal to have a hardened spindle ride against it for decades, provided it has oil. ACME shafts wear from
hard silica bearing dust contaminants riding in oil. Both the nut and the ACME screw will wear. It becomes a (thin) lapping compound. The abrasive embeds into the softer metal, and the truly rock hard points cut away at the hardened steel. Such a lap will wear anything, even glass.
I once made up a "bronze epoxy" ACME compound nut repair from a mix of JB-Weld and powdered bronze, using the ACME screw with release agent (wax polish) as a mold. The wear in the middle was such that I could not undo it past the unworn ends without cracking it. I had to lap the last one to get a working version. If the nut is worn, then do also critically examine the ACME screw. Getting up a new nut may only leave you with one that is OK, or a bit tight, near either end of the screw, but still somewhat sloppy in the middle.
Here is a near worst case, crappy condition case of worn nut with a split crack in it, before restoration attempts.
View attachment SB9-Compound Acme Nut2.mP4
Here is a picture of the mold getup, made from some ex-HDPE water pipe.
The bronze-loaded JB-Weld was a reasonably fast fix, but I consider it temporary. This last image is of a trial test nut, with a bit ground off to show that it is mostly bronze. It was interesting, and got me out of a problem at low cost, meaning "quickly", but I would not do it again.
For you, you might fix everything to be fully likeable if you discover the majority the slop only needed some attention to the thrust spacing or adjustments. If it is wear, then a dandy project is to get up a new nut and ACME screw. Even if you get a nut, and adapt it your lathe, or start from a piece of bronze bar, and buy a ACME tap, or go the whole way, and cut internal ACME. Whichever route you take, the payoff gain is big. Having a fully fixed up lathe, with just enough backlash to let you know where the the measure point is, will put a smile on you!