In many cases, an update to your carrier’s voicemail app or settings can resolve the issue, but don’t forget to call your voicemail number to check if it is set up correctly. Once you’ve set up your voicemail, you’re free to switch off when you need to. There are other ways you can stay in contact, however.
Besides creating an audio file in the right format (Adaptive Multi-Rate), you also need to trick the iPhone to upload your new greeting to the AT&T servers:

To change the number of times your phone rings before voicemail answers, please contact us and a representative will set your ring cycle.
If you had a generic voicemail greeting and recorded a personalized voicemail greeting you can change to the custom voicemail greeting by saving your custom voicemail recording. You should first listen to your recording by playing it from the menu and can then choose to keep the custom greeting or revert to the default one. Don’t forget to confirm your changes by tapping “Save” when done. How do I change my voicemail message on Samsung?How do I delete my voicemail greeting on Android?How do I change the default voicemail on my Android?How do I manage my voicemail on Android?How do I change my voicemail greeting?How do you use voicemail on Samsung?How do I remove a name from my voicemail greeting?How do I turn off my voicemail message?How do I turn off voicemail on Samsung?How does voicemail work on Android?What is my voicemail PIN?What is the best voicemail app for Android?How do you unlock voicemail on Android?Is there a voicemail app for Android?Does Samsung have a voicemail app? On Android devices above Android 5 (Lollipop), open the Phone app. Then, press and hold “1” to call your voicemail. Now, enter your PIN and press “#”. Press “*” for the menu. Press “4” to change settings. Press “1” to change your greeting. Tap the three dots (upper right corner of screen) Tap “settings” Tap “voicemail” Tap “advanced settings” Tap “setup. Tap “voicemail number. Enter your 10-digit phone number and Tap “OK. Tap the home key to return to the main menu. Open your device’s Phone app ​. At the top right, tap More . Tap Settings. Voicemail. You can: Change what handles your voicemails: Tap Advanced Service. Set up your voice mailbox: Tap Advanced Setup. Change your notification settings: Tap Notifications. Open the Google Voice app . At the top left, tap Menu Settings. In the Voicemail section, tap Voicemail greeting. Next to the greeting you want to use, tap More Set as active.