Very interesting thread!
Good points about the tuning and chord shapes 70sSanO. It probably depends also on whether you mostly strum on the uke or also do fingerpicking, in which case I would imagine you could translate some of your fingerpicking abilities on many other instruments with different tunings and 3-4 strings. I actually find it quite fun to experiment with different tunings on the uke to get other flavours from it than the my-dog-has-fleas one, but I mostly try open ones and then do some hammer-ons, pull-outs and fingerpicking, or clawhammer (I'm not very good at clawhammer though); no actual chords (because it would probably fry my brain haha). It's a lot of fun!
One big leap would be to play a similar string instrument but with no frets. Almost guaranteed to miss the right note unless you've practiced for years and years (but no fret buzz hehe).
+1 on the post about ukulele vs. percussions: I play samba drums (surdo and caixa) and a bit of drum kit, and the rhythmical skills you get from playing it translate very well to the uke and vice versa, as the uke is very much a rhythmic instrument itself.
Actually, one fun little tick I recommend is to drum simple rhythm patterns with your index finger on the soundboard in between notes while fingerpicking - it's something Malian ngoni players (and Senegalese/Mauritanian xalam/hoddu players) do and it sounds really nice.
Ah, the ngoni! The king of all string instruments imo. I wish I could play it even a tiny bit, but it's really hard, you have to be ultra-precise and have a really sharp ear (no frets... even different tunings for different types of songs...).
I agree with what someone was saying earlier about not spreading yourself too thin by playing too many instruments, but I do think it's a good thing to try playing (or even just learn about) other similar string instruments with different music traditions and playing techniques in order to expand your musical horizon, and then try applying what you've learned back on the uke. It can only make your playing more versatile