iOS 15 verification failed because you’re no longer connected to the internet

Try first to optimize the range of the mobile network, see this Apple article:

If you see No Service or Searching on your iPhone or iPad

If you see No Service or Searching on your iPhone or iPad - Apple Support

See the following Apple articles:

Update your iPhone, iPad, or iPod touch

Update your iPhone, iPad, or iPod touch - Apple Support (IN)

If your iPhone, iPad or iPod touch won’t update

If your iPhone, iPad, or iPod touch won't update - Apple Support

You can do the update via PC, see this Apple article: Update software on iPhone, iPad or iPod touch in iTunes on PC

Update software on iPhone, iPad or iPod touch in iTunes on PC – Apple Support (UK)

If you have a Mac see this Apple article: Update your iPhone, iPad, or iPod touch with the Finder

Update your iPhone, iPad, or iPod touch with the Finder - Apple Support

You need about 7 GB device storage and good Wi-Fi.

If they are On, turn Off Hotspot & VPN.