Q: My phone won't update

Since you just changed internet providers recently, let's start with the basics:

Connect to Wi-Fi on your iPhone, iPad, or iPod touch - Apple Support

Then go to one of the links at the bottom of that article:

Learn more

• Learn how to join a subscription Wi-Fi network.
• Get help if you can’t connect to a Wi-Fi network on your iPhone, iPad, or iPod touch.
• Learn what to do if you can’t turn on Wi-Fi because the setting is grayed out or dim.
• Learn more about settings for Wi-Fi routers and access points.

Also make certain that your iPhone is not serving as an internet hotspot for your computer when it is plugged in. If you are reaching the stage after it has Extracted the software and is Preparing the iPhone, the connection will be lost as the iPhone boots down to prep for the new software to load. In the installation process, which it is pulling from a file which has downloaded from Apple's servers to your computer, it makes an attempt to contact Apple once again to verify the certificate on this install and if that internet connection is not present because your computer is searching for its lost connection to your iPhone as a resource, this will happen.

To solve for this, make sure Personal Hotspot is turned off on your iPhone in Settings>Personal Hotspot and try again. Also make sure your computer has internet access when this is done.