Unable to install iOS 15.1 update

Assuming you're trying to update your device wirelessly, and bearing in mind that to perform such a procedure you need to connect your device to a reliable and fast Wifi Network with internet access (no cellular data, unless you use a 5G network, no personal hotspot on your device, no VPN connections are admitted, and I suggest that you keep the device plugged into power during the whole update process), read this suppport article:--> If your iPhone, iPad or iPod touch won’t update

If the issue persists, try to update it via computer (if you don't have a computer borrow one from your friends/neighbors, or ask for help at an Apple Store or Authorized Service Provider):

Learn how to update your device manually if you're using a Mac with macOS Catalina or later, or if you're using a Mac with macOS Mojave or earlier or a Windows PC.

If you still get no help, contact Apple Support directly, using this link:--> Get Support

Regards

Giulio

Well are you? The error states that you're no longer connected to the internet - have you verified your connection?