Thanks, Lorne.
This was a weird one. I now understand what went wrong. Before changing all the phones globally, I wanted to test this on one phone. So I manually programmed the message key using the web interface for the phone, and before I got a chance to test it, the Endpoint Manager automatically reprovisioned the phone, as it should every 24 hours. So I was actually sending *97 rather than *98237. When I reprogrammed the key, and then immediately test it, it worked correctly.
Now that I know that it works, what I want to do is program this automatically on all the phones using the Endpoint Manager, as suggested in Shared Voicemail Boxes .
On the Snom 821, the basefile contains <dkey_retrieve perm="">speed *97</dkey_retrieve>, so it looks like I need to change it to <dkey_retrieve perm="">speed __voicemailExt__</dkey_retrieve> .