This is what's left dangling from the ceiling:
(Sorry it's so blurry - I had to take it one handed in the dark while using my other hand to point the light fitting in the right direction.)
This was attached to the light bulb, i.e. it used to be part of the black bit at the bottom of the ceiling fitting, but it came away in a shower of small bits of black plastic:
These are the things I took off afterwards:
The basic idea is that you would start with the ceiling thing (including the bit that the bulb screws into). Then you put on the lampshade, and then you screw in the white bit, to be the bottom part of the "egg" fitting and hold the lampshade in place. Finally, you screw in the lightbulb.
Based on my hazy memories of last time, I think I'll need to take out the black bit that's still dangling from the ceiling and replace it with a new component which includes the metal screwy bit. After that, re-attaching the lampshade/white bit and the new bulb is all easy.
Taking another look at the dangling fitting, the white dome bit on top unscrews and can be slid up the cable. That then reveals the rest of the black bit:
It's not very clear from that picture, but the white electric cable contains two inner cables; presumably blue = neutral and brown = live. Where they join the black bit, the coloured plastic is stripped away, and the actual (copper?) wire inside is presumably attached to some kind of terminals, as in the plug for an electrical appliance. The two plastic cables both meet the black bit in roughly the same place (on the "horizon" of the circle as seen from above), but then they branch off in opposite directions along the perpendicular diameter.
So, I'd guess that I can somehow detach the blue and brown cables, thus freeing up the black bit which I can then throw in the bin. I would then need to get hold of a new black bit and reattach those wires. It's probably prudent to turn off the fuse for the lighting circuit rather than just relying on the light switch being off in this room, at least for my first attempt. I think this would also be easiest in daylight, rather than having to balance torches at strategic locations in the room.
Anyway, if my blog goes strangely silent, then let this be my epitath: "How hard could it possibly be?"
Edit: On closer inspection (looking down on the light fitting from above), there are two small screws which are holding the copper wires in place, so it should be pretty simple: just unscrew them, remove the wires, then screw the wires into the new fitting. The only snag is that white circular piece of plastic; it's in the way, and I'm not sure what purpose it solves. I also can't see any obvious indication of which wire is supposed to go into which terminal, so there may be an element of trial and error on the new fitting.
Edit 2: On further reading, it seems as if all light fittings are supposed to include an earth wire nowadays. In other words, it's a bit dodgy for mine to only have live and neutral, implying that the wiring is very old, so this may be a bigger job than I thought.