what uke is it? (that will give us a guide). you should also bear in mind that high action (you say you have lowered it) is not necessarily the issue if you have bad intonation.
Going sharp at the nut is often symptom of nut being too high. Does it go sharp when you press at first fret, but less so if you fret lightly at the first? If so - nut slots need to be deeper - but be careful doing this yourself!
Bad intonation down the neck can be down to a whole host of things, most seriously with badly set frets, or slightly less seriously with a bad saddle (though very badly with a duff bridge position)
Taking things step by step, a change of string type can help - what are you strung with?
If you bear in mind, say, and electric guitar - the individual saddles are adjustable, and often need tweaking if you change from heavy to light strings.
That isnt possible on a Uke, but if you are on heavy thick strings, you could try changing to lighter type.
Also, sometimes you can get bad strings that have been manufactured oddly- a string change, or even swapping the strings around so you put the tuner end at the bridge can sometimes help.
Very cheap ukes (such as my £15 Mahalo) have truly dreadful intonation. I have the action set spot on, and have tried different strings, but quite frankly the bridge and / or the frets are just plain wrongly set. For £15 I am happy to chalk this to experience and try to tune mid way between good and bad to strike a balance. That said - if my £200 Mainland had bad intonation, I'd be sending it back!
