You should use purpose made nut files. Get them in appropriate dimensions from Stewart MacDonald. You can rock them from side to side a bit to make a larger slot than the nominal size.
The action at the nut should feel much like it would if you capoed the uke at the first fret and played it. In other words, the action should be like it's coming off of a fret. It's REALLY easy to file too far. Take your time.
Good action at the nut will even make the action noticeably better way up the neck.
Both because it's easy to go too far, and also to account for any wear over time (albiet much less on a uke than a steel string guitar), i like to have the string height at the nut just a little more than when compared to a fretted string.
A lot of ukes have an ebony nut also,.....which seems to me could wear a little faster than a bone or ivory one. So i've found adding maybe .3mm or .4mm more than the string clearence gets when fretting the 1st fret is a good place to shoot for. Remember, it's a lot easier to take off height than add it. Having to make (or pay for) a new nut because you filed a few too many strokes is not fun.
... the very slightly higher nut action will not be noticed anyway, once you get the really "extra" nut height off. I've had a few that were 2 to 3 times heigher than optimum string height at the nut,......unplayable. Besides the much harder fretting at the lower fret positions,.....all the notes are noticably sharp.
BTW, i was a bench jeweler for 35 years, (repairman and diamond setter) and used to using small files. I got along fine with 2 small tapering rat tail files for most of my nut slot work. One was coarser and used to get close, with the finer file finishing up. You had to taper the nut toward the headstock sometimes with a sanding block, because a wide nut can't be slotted with a tapered round needle file. ( the taper catches before you get to the far side of the slot)
......but in finishing up my recent Cordoba 25CK adjustment i was day-dreaming a bit and snapped the tip off of the finer file! Looks like i'll be getting a set of nut files at 55 years old! :drool: