The problem was the bridge/saddle placement. Elementary physics says that to go up an octave, you shorten the string by half. On a string instrument you actually want to compensate also for the tightening of the string when you press on the fret, so you want the 12th fret a little closer to the nut than to the bridge.
On this instrument, the distance from nut to 12th fret was 177mm, from the 12th fret to the saddle, 166mm. No wonder the instrument's intonation was out!
Using steam, I loosened the glue attaching the bridge, and worked a thin blunt blade between bridge and body. Eventually I could remove the whole piece.
Another problem was that the action is too high near the 12th fret. You want a little less than three millimetres at the 12th fret — this instrument had 7mm. Taking 4mm off the saddle was too much — it would sit below the slot which holds it! — so I sanded a couple of millimetres off the bottom of the bridge before gluing it back. I actually got the position slightly too close even then, but it's too much trouble to pull it off again.
I replaced the horrible nylon strings that the instrument came with some decent ukelele strings (made by Aquila) and now have a playable instrument.
I also used a fine fretsaw to deepen the notches in the nut to lower the action.
It doesn't look very good though, until I can find some brown paint to cover where the bridge used to be.
Another problem with this instrument is that the knick where headstock slopes away from the neck to hold the tuners is not very pronounced. As such, the strings can float way over the nut, unless one is pretty careful in pushing them to the bottom of the capstan on each machine head.
It's a bit silly that this instrument, with good quality machine heads, and a reasonable pickup, has such flaws.
Although playable now, I upgraded to a decent instrument from Rocky Creek Strings