On my return I soldered the original cap back on (a fiddly job to get it accurately centred - past experience with soldering surface mount ICs came in handy here). Once I'd cleaned off the excess flux and solder and given it a polish you can't tell that it has been re-attached. It looks better in reality than in the photo (very tricky to get good in-focus photos of such small objects without a macro lens).
After putting the repaired key back in, I had a close look at all the others to see if any of the other caps looked like they might be coming loose. It turns out there is at least one more that has been replaced that I hadn't spotted before - I thought it was just very dirty, but it's actually made of rusty chrome-plated steel (plus it's thicker and a slightly different shape). I do wonder why so many of the key caps have fallen off (seven that I know of). If they were dry joints to start with, I would have expected them to come off within a few days of starting to play the instrument at most. If they were good joints, I would expect them to stay on pretty much forever.