Not a doctor nor do I play one on TV - that said, if you don't have a medical predisposition toward joint/muscle problems then playing the ukulele, even for hours a day, should not cause any problems if you are not abusing joints by playing in awkward positions. Yes, some guitarists have developed RMI issues after decades of playing - but they are playing instruments with far higher tension than our ukuleles have and there is no way to know what medical predispositions may have applied.
So, how do you know if you're abusing joints? Basically, if you work up to it gradually so your muscles come up to strength without being overtaxed and your fingertips harden up, you should be able to play for hours without discomfort. If something is uncomfortable beyond just a little tenderness in the fingertips of your fretting hand then you need to look at improving your posture and technique and/or cutting back on playing.
Also, remember that your entire body is connected. You can cause wrist problems by having the wrong posture in your back, and so on.
John