iPhone shows “network update failure,” but I have the latest software

There are two major Updates in an iPhone / iPad:

1. iOS Update (Recommended): Released by Apple Inc basically for improvements, security and bug fixes. My 10 years of experience says, it has no affect on Data, Calls & SMSs if not updated
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2. Carrier Update (Required): Released by your Cellular Service provider (Carrier Update), affects your Outgoing / Incoming calls, SMS, Data etc..
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How to Update Carrier Settings

You can manually check for and install a carrier settings update with these steps: Make sure your device is connected to a Wi-Fi or cellular network. Tap Settings > General > About. If an update is available, you'll see an option to update your carrier settings.

Update your carrier settings on your iPhone or iPad - Apple Support

How to update iOS

Go to Settings > General > Software Update > Automatic Updates, then turn off AutomaticUpdates. At any time, you can check for and install software updates. Go to Settings > General > Software Update. The screen shows the currently installed version of iOS and whether an update is available.

Update iOS software on iPhone - Apple Support