I use a 5mm fingerboard.

The neck (ie, not the fingerboard) is about:
12-13ish mm under the nut (so 17ish mm with FB),
15-16ish mm around the 12th fret (so 20-21ish mm with FB)

YOu can make a uke neck way thinner (due to very low string tension, but a too thin neck is way more uncomfortable than a too thick neck.