I had another go at this yesterday (after buying another mic), and it still doesn't work on my 950 (running WM10), BUT I happened to have my old 920 (running WP8.1) on the desk and - on a whim - tried with that. What do you know, it works perfectly!
After doing a bit of research into headset connectors (eg
this thread by alv3st3r), I wondered if it was caused by the 950 and the microphone being wired for different standards (the 920 has a special universal audio jack) and ordered an AHJ-OMTP adaptor, which just arrived. Excitedly plugged it in and... nothing :-( I've also tried the mic (with and without adaptor) on an 830 (running WM10), and that didn't work either.
So, either (1) there's something special about the 920, (2) external mics don't work on WM10 (regardless of wiring), or (3) some combination of the two. After I've recorded the next video (to test the quality of the 920) I might upgrade it to WM10 and see if anything changes, just to rule that out.