Well, I've finally had to sell my Apple watch because of this. I did ask Apple to return it but because I'd had it too long (5 months) they wouldn't take it back which is fair enough. AppleCare were as helpful as they could be though they seemed more to think it was some allergy causing a reaction but it wasn't those sorts of symptoms.
The pain/numbness/tingling in my wrist, fingers and hand gradually got worse to where I couldn't wear it. Eventually as a test I didn't wear the watch for a fortnight and in that fortnight I had no problems at all (and also wore my normal watch). After 2 weeks I put the Apple watch back on and the pain came back almost straight away, certainly within an hour of wearing it.
I had tried 3 different straps (nylon, milanese loop and a 3rd party metal link one) and the watch was definitely not too tight so it is some aspect of the watch itself.
I think this is either:
1) the domed shape of the heart rate sensor pushing down into my wrist and causing pressure on a nerve, Apple Watch does not sit on the wrist in the same way as a normal flat backed watch
2) the heart rate sensor itself interfering with my body somehow
3) one of the radios in the device (such as Bluetooth)
I wasn't aware you could turn off the heart rate sensor in the settings, I now wish I had known this before I sold it so I could give that a go to rule it out. There are people on the fitbit forums having similar issues with fitbit devices and some people have reported that turning off the heart rate sensor solved their problems. Other people have been incredulous at this and said 'it's just a green LED' but this is all new technology so it's plausible to me that some people might be affected in unknown ways.
Similarly other people have reported that turning off the bluetooth solved their problems on a fitbit. There would be no use having an Apple watch if you couldn't turn on the Bluetooth but I could certainly live with no heart rate sensor.
It seems clear to me that these issues affect only a small number of people (for whatever biological reason), but enough people are experiencing this to where I think there is definitely some issue that certain people are sensitive to. It would be good to get this narrowed down.
Personally I am now wondering about trying another watch like a Samsung Gear S3 to see if I have the same issues. It's a gamble but they do at least have a flat back I think so I could rule out one of those three things.