Why is there a growing time lag in gaining access to voicemails - I’m talkin’’ hours.

Voicemail is stored on the carrier’s voicemail server as soon as it is recorded. The carrier’s voicemail server then forwards it over cellular data to your iPhone when it has a cellular data connection. You can verify this by calling your carrier’s server and logging in with your voicemail password; for each message it will say when it was recorded. If it hasn’t been recorded on the voicemail server that is a carrier problem that has nothing to do with your phone. If it doesn’t have a connection when the message is received it can’t send it. But it doesn’t try continuously; I don’t know the schedule it follows, but it tries periodically after that.

My visual voicemail arrives within seconds of the time it is recorded, every time. That is, I see a missed call, and a minute or two later I see a “badge” on the Voicemail icon in the Phone app and I can listen to the voicemail. I may have to wait a few minutes for the phone to transcribe the voice message into text, but the recording is there right away.

Hello Steven Weignberg1, The best thing you can do is contact AT&T your carrier and ask for assistance on your Voice Mail. I have AT&T and don't have this wait time for a Voice Mail. If someone leaves you a voice mail it should show up instantly.

GeeZ06 is correct. Note as a possible reason that you must have a cellular data connection for visual voicemail to update. It cannot be updated over Wi-Fi.