More times than not, if it's sharp at the first fret it is the nut. If it's sharp on down the board, it can be the saddle. Maybe your instrument could use some work on both sides of things.
Someone mentioned not pressing too hard. Good call. I've been playing stringed instruments for over 20 years and I STILL have to be mindful of this sometimes with ukes especially. Lower tension strings are A LOT easier to pull sharp simply by pressing too hard. You have to develop some great touch to keep that intonation in check.
There is one other possibility, which is the worst scenario. That would be that the bridge on your uke is located in the wrong place. Considering it seems like the intonation is consistently off, this is possible. It'd be ideal if you could take that bridge and move it back about 1/16" of an inch and test that. Of course that isn't possible with fixed bridges.
There is one other solution to that that can be done in addition to lowering the action, OR simply in place of lowering the action if you like the current string height.
That would be to sand the top of the saddle so the strings break point moves back some.
You'd need to remove the saddle and sand the top side that would be closest to the headstock at around a 45 degree angle, as a good starting point. Then round over the top. The thickness of the saddle will determine how much you can move the break point back. I usually just tape a piece of sandpaper to a flat table and then hold the saddle and run it back and forth over the paper. Don't apply a ton of pressure, let the paper do its job.
Don't forget to round over the top a bit after finishing. I'd suggest a super fine grit for the very top after finishing. You want that to be as slick as it can be. I usually just hold the saddle and work the paper carefully back and forth over the top and sides near the top with a motion that rounds it up. It usually doesn't take much.
I'll attach a little crude drawing I just did of this in the Windows Paint program, because I'm not sure if I'm doing a great job explaining it with words. Maybe it'll help.
A little thing that helps me remember which way to move a saddle to compensate for intonation problems is: Flat is forward, sharp is back. I remember that flat needs to move the saddle forward since "flat" and "forward" both start with an "f". ha. By "forward" I mean towards the headstock. So sharp, like in your case, is back. Maybe a silly little thing there, but maybe someone else out there thinks like me and that'll help.
Click on the pic and it should enlarge:
View attachment 95599
Keep in mind, as some others have already noted, switching to a different string set can change all these factors. Ideally, an instrument is setup with a particular string set in mind and you stay with it. I think that was worth repeating. Also, as strings age things change a bit too.