I imagined the webs on the slotted piece being on the underside. ie the flat part with the slot being the top. The circular tube being to locate the part on the lower shaft.
Hadn't thought of that. It makes sense.
There are quite a few broken bolts, but no worn or broken flanges etc. My gut feeling is that the drive pin has fractured, but the current collection of bits will work with a pin shorter than it currently is.
As you say, the next job is to open the sluice and spin the wheel. From there it should be quick enough to fit the 'new' coupling, the tyre and bolt it up.
It is an exciting project, however, it takes 2nd place to getting a house comfortable. It was it a real state, having been abandoned for many years. It's in good shape now, but plenty still to do.
The mill is in France, not far from Lourdes. Despite being close to a main road, there are no services near by. IIRC, the cost for a small supply was around €60k (after a substantial grant) and more if you want to inject. That gets the supply to the boundary, then there would be the costs for 100m or so to the mill.
There is what looks like a sluice control system on the wall. However, there are no actuators, just limit/open/close switches. If restored, this could control the frequency, but manual tweaking will be required to see how the system copes under load. The current electrical installation is basic, so it's not obvious what was previously connected. There are a lot of halogen lights around.
If I can drag my friend away from building, I'll go back over and we'll spend some time trying to get it spinning.