In a conventional instrument, the nut performs three main functions. It determines how the strings are spaced apart; it defines that end of the scale length; the depth of the slots can be adjusted (in combination with the saddle) to provide the preferred "action", or height of the strings over the frets. A zero fret takes over two of those jobs. It is the "stop" for that end of the scale length, and it fixes the action at that end of the string. That leaves the nut with only one job to do:- it is there simply to space the strings across the fret board. I've made a couple of zero fret ukes with no nut at all. You just have to find another way to space the strings.
You cannot adjust the action so easily with a zero fret, but that is not a problem. The first time I used a zero fret, I set it a little higher than the other twelve frets, but subsequently I made them all the same height, and that works just fine for me.
Hope this helps.
John Colter.