OK, I just filed the bottom of the saddle a bit on sandpaper.
The C-string is very sharp at D (about 25 cent) and extremely sharp at C+8va (50 cent). Prior to this exercise it was around 20 and 25 respectively. I think I'm going to try replacing the string (great... it's only been a week) and try again. Back to Aquila from Worth, I suppose... (and the one white string in four that comes with it).
It is possible that you have a damaged string - a bad spot in a string can throw off your intonation suddenly at some point and no amount of "setup" will correct it. However, if changing the string doesn't help then keep reading...
You're dealing with two separate issues, actually. The string being sharp when fretted at the first and second frets is pretty much always (barring a bad string) the fault of the slot in the nut not being deep and/or wide enough. This is usually most noticeable on the C-string because it is the thickest string and the nuts usually have molded-in slots all the same size so the C string is the most likely to be sitting too high. However, it can, and often does, happen on all of the strings.
Basically, you should fix one thing at a time, and
start with the nut. Work on each nut slot until its string can be fingered (gently, just enough pressure for the note to ring clearly and fingered just behind the fret) without the note going sharp. I.e. when this is part of the setup is correct the note at the first fret will be C#, not C# plus a few cents. It is
always possible to achieve this without buzzing
unless the fretwork is bad, the neck back bowed, or the bridge saddle has been lowered too much.
If you have a string start buzzing before you've achieved good intonation at the first fret the next step is to either dress the frets or user super glue to put a little height back into the nut slot. Which one decides to do is up to the individual and how much tolerance they have for out-of-tune notes.
Once you have good intonation at the first fret, it's time to start looking at intonation up the neck. Saddle height makes some difference, but has nowhere near the impact on intonation up the neck that the nut has on intonation at the first couple of frets. Usually, adjusting intonation up the neck requires first finding the strings that intonate the best on that uke, and then if necessary making a compensated saddle to fine-tune it. Up the neck the goal sometimes has to be to make the intonation consistent across the strings even if it isn't exactly correct. I.e. if all of the strings are 5 to 7 cents sharp at the 12th then chords up there will still sound decent and most people won't notice. If some are five cents sharp and others five cents flat you have a problem...
With your one string being so far off chords are going to be pretty bad by the fifth fret, even if you get everything good at the nut.
John