I have AT&T service. After updating from 9.0.1 to 9.0.2 I lost the use of visual voicemail. When I touched the voicemail icon in the phone app it would start an audio call to voicemail. I spoke with Apple for a half hour and they tried many things, none of which worked.
One of the items was to change the voicemail password. I did that with settings>phone>change voicemail password, and the process never completed - I was left with a screen that said "Updating password..." but it never cleared. I then called voicemail by touching the voicemail icon, and followed the menu to change password. That also failed to resolve the issue.
However, when I called 611 and followed the menu to reset my password, and received a text with a temporary password, the visual voicemail immediately worked, as did the change password option in settings.