12th fret height - between 2mm and 3mm (.080 to .120 inches). 3mm is best for those who mainly strum chords on the first five frets, 2.5 mm (.100) is a decent all-round compromise, 2mm is for the player of fast melodic runs.
Treat bridge and saddle as a single unit. You want a certain minimum height above the soundboard to the peak of the saddle, otherwise your uke will be quiet and dull-sounding. I aim at 10-12 mm (roughly .4 to .5 inches). If you've built it so the total height it much lower then you should consider whether you can fix that, e.g. by re-setting your neck or putting a tapered whim under the fingerboard. But 8mm (0.32 inches) can still work, less would worry me.
Then your bridge needs to (a) hold the string ends, and (b) hold the saddle. The lower it is, the harder these are to achieve. You also need the strings to break downwards from the saddle peak at 15 degrees or more. So you want at least 2mm (0.8) above the bridge I'd say.
Suppose you are forced to go as low as 6mm (0.25 inches). Your uke might work, but this leaves your bridge at 4mm which is too shallow to cut a saddle slot easily, I'd say. The last ditch solution is to make the bridge and saddle as a single piece and to attach the strings by drilling vertically through the bridge. Then you feed the string through, fish it out through the sound hole, and knot a small bead on the end. Works very well.
It's what I did here, though in this case for the look rather than because I'd gone too shallow:
