iPhone 7 iOS 12 to iOS 15.8.5 OTA Update Fails: ‘Not Connected to Internet’ Even on Wi-Fi

If you have verifying update it sounds like it has downloaded.

Look at what may be blocking connection like have you installed VPN - uninstall it

You can also try to delete the download and try again.

If your iPhone or iPad won't update – Apple Support (UK)

1. Go to Settings > General > [Device name] Storage.
2. Find the update in the list of apps.
3. Tap the update, then tap Delete Update.
4. Go to Settings > General > Software Update and download the latest update.

At this point, virtually all iPhones that are running iOS 12 are iPhone 5s, 6, or 6 models that cannot update past the latest release of iOS 12. The iPhone 7 would, at one time, have been eligible to receive iOS 12, so I suppose it is not out of the realm of possibility that you let it update that far, then blocked further updates.

If you did that, then to get to iOS 15, you would likely need the aid of a Mac or PC.

Update your iPhone or iPad - Apple Support

"… you can use your computer to update your device. Learn how to update your device if you're using a Mac with macOS Catalina or later. Or, learn how to update your device if you're using a Mac with macOS Mojave or earlier or a Windows PC."

You say that "I don’t have access to a Mac or PC to manually update, so I can’t use Finder/iTunes." Given that, you may be stuck with iOS 12.