5. When you're ready to begin recording, tap on "Record" and begin speaking. Once you begin recording, you'll see the line turn red. This indicates how much time you have left.
One of the best and quick fix to this issue is to turn off Wi-Fi for once and switch to Cellular data connection and then try to change the default greeting to custom on iPhone.

6. After the message has played, you’ll be given some options for what to do with the message e.g. delete or save it, etc. along with the corresponding number to press.
If you’ve already created a custom voicemail on your iPhone, you can tap Play to see how your voicemail sounds. If you’re interested in creating a new message, tap the Custom button. From here, you can start recording your message by tapping Record and tapping Stop whenever you’re finished. Once you’re finished recording, tap on Play to make sure the message comes across as you’d like and hit Save (in the upper right) to save it as your voicemail.