Can you please provide a little more detail. For voicemail set up on my iPhone SE, I ihad to initially switch/set Cellular data to ON. Once setup I can switch cellular data to OFF and voicemail works OK. It would be great if you can please update this post with your solution. Thanks.
"Cellular data has to be turned on and left on in order for voicemail to work on my SE" - You are right. For Voicemail notification and visual voice mail to work, cellular data needs to be ON. If Cellular data is OFF caller is able to leave a voice mail message which can be retrieved by dialing voice mail.