as a classical guitarist I can tell you what works for me.
Your best bet is to free up both hands so you can move around easily both on the fretboard and to strum or pick.
while I can play without a strap as you can see on some of my youtube videos I prefer to have strap buttons installed on most of my ukuleles. there are other alternatives including the Uke leash which gives some added support as well.
once you start using a strap you now have both hands free. And for the left hand specifically I try to use my thumb only for balance not support. The optimum is to have the pad of the thumb resting behind the neck, balancing your hand so your fingers can move around. it may go up and be seen over the fretboard but only because of the size of the neck, not because it's supporting the uke.
Now without a strap you do have to kind of juggle between proper (free hands) technique and providing some support to the uke, although the main support in that case is from craddling the uke on the crook of your right elbow or holding pressure on it with your inner elbow against your mid-section.
In any case find what works for you, and stay with it consistently...
as a matter of fact one of the Assad Brothers, world re-known classical guitarists plays holding a classical guitar with his elbow pressure as I just described. I don' tknow how but I saw him play for 90 minute like that standing up, and with incredible technique and musicality, so I guess anything is possible. then again he's been playing for 40+ years!
Good luck.