Let's make sure that voicemail is working as it should. When did the delay with retrieving your voicemail begin? Unfortunately the Visual Voicemail on the iPhone is an embedded feature created by Apple so we cannot remove it. However, if you prefer, you should still be able to access your voicemail by dialing *86 from your device.
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I sometimes need to screen calls and two from the last week left messages, but I didn't get the VM alert until later. The first one came in nearly an hour after the call and the second was 30 minutes after the call. Apple Support said this is a beta feature and the only way to disable it, is to have our service provider do that . . . but it sounds like I'm stuck with it.