My Rockwell Delta 10" is just like that, however when you wind it to it's extremities of travel the backlash disappears which points to the lead screw rather than the nut being the culprit. Make sure you source the right bit :O)

You are not looking in the right area. Look instead at the handwheel end of the cross slide screw. Unscrew the chromed slotted 'nut' (anticlockwise with something that doesn't damage the slot) Now undo the hex socket grubscrew that holds the handwheel to the cross slide screw. Now fit the handwheel to remove all slack, tighten the grub screw Refit the slotted nut.

As for spares, you are out of luck for that area of the lathe because that type of cross slide / dial was discontinued in the early 1960's

Part 2

Here's a useful diagram but it is really annoying that they don't tell you the threads of fixings.
It also shows the more advanced modification of fitting thrust bearings which they did on later models, but they didn't do the top hat.
Next we go to the cross-slide nut. Notice it has a cylindrical mounting boss that moves in a hole. If this is a sloppy fit then it will contribute backlash. It has a grubscrew 155 which pushes a brass slug 156 (or ball bearing) sideways against the side of the hole to lock it. Contrary to the diagram the slug should face the operator. I will leave it as an exercise for the reader to work out why.

That's it for adjustment. The nut is not split for taking up wear. The screw itself will become worn in the middle over time so that provides another source of problems.

Southbend did expect there to be backlash and if you read their instruction manual " know your lathe" there is an instruction to set the tool height above centre line contrary to the normal instruction, This upsets the purists but is very sensible on all lathes when done with some thought. Above centre there is a component of force on the tool outward which ensures the backlash is taken up.

Part 3 but on other topics a Boxford owner needs to know.

The screw in the middle of the star wheel that engages the feed clutch is******* left hand thread********.

The grubscrew on the right of the apron that holds the halfnut engagement lever in place has another gubscrew deep down the hole,

The backgear lever on the front has a brass slipper running in a groove in the bull wheel. Check this isn't worn as the pin it pivots on will score the groove. Oil the groove even if not using backgear.

The clutch is not there to let you walk away and let the saddle thump the saddle stop which will move if you do. You adjust it to just not slip and keep your hand on the feedwheel letting it run through your fingers. As you come close to the stop you grip the wheel and slow the saddle slipping the clutch so it gently touches the stop.

Don't use grease or motorbike chain oil on changewheels as it holds dust and swarf. Use normal thin oil and use it occasionally to wash off the crud. Ditto in the apron, QCGB and leadscrews. Way oil for the ways. The apron probably leaks so put a dish under it.
Space changewheels with a piece of paper between the teeth to minimise noise.

There are a large number of spare parts available in the US for South Bend lathes, which your Boxford is based on.

You might be wise to join Practical Machinist website in USA who have a sub forum for South Bend. The guys there can probably answer any questions you may have.

I was going to reply, but Bazyle has said everything and more perfectly. +1. Keep the grease for your Harleys, it has no place on or near a lathe. Nice Shovel in the background by the way….

Yeah i know, very very nice.

On my ME10, the dials are friction loaded. Under the grub-screw is a small spring and a brass disc pressing on the shaft.The friction pressure is controlled by adjusting the grub-screw,(hope my memory is correct).

Sounds as if your grub-screws are over-tight. Try slackening them. It's also possible that your springs and pads are missing.

Hope that helps,

Clive