I've seen this issue reported many times over the years, and lots of "things to try", but never ever have I seen a user say that any of them worked. I too have tried resetting network settings, ensuring that mobile data is turned on, and verified that the problem happens whether I'm connected to wifi or not.
I've been with Verizon for over a decade. This is the most ridiculous situation I've ever seen- getting my voicemail is EXTREMELY IMPORTANT. People are losing out on job interviews, extra shifts at work, client appointments. It doesn't help me to get a voice mail notification hours or days after the message was left! YOU NEED TO FIX THIS PROBLEM!!!!
We know the importance of having access to your voicemails in real time. We welcome the opportunity to address your inquiry. Currently, the issue with delayed voicemail notifications is a known issue. It is occurring with multiple iOS devices across multiple wireless carriers as shown here: http://spr.ly/6609EItil. Please know that the issue is under investigation and we are working with Apple for a software fix in the near future.