You may need to connect the device to iTunes and update/restore it from there. See iTunes help and search for restore or follow this link - https://support.apple.com/en-us/HT205239.
The above article covers your specific "slide to upgrade" issue.
The good news is that you should be able to connect your phone to iTunes and make a back up even in the state that it's in. There are only three steps in the article above, so definitely try that first.
If you can't get iTunes to trust your device, then you might call AppleCare to see if there's any other option. Since Apple acknowledges that this issue will be fixed in the future version of iOS, they can best advise you how long to wait or if they can help you back up your device.
AppleCare has training in and can certainly guide you in searching for potential backups.
- contacts and calendar data stored / synced online
- iCloud backups
- computer backups or syncs
- data that's stored in iCloud and that will re-apply to the device once you wipe it and render it operational
Here are the wipe instructions if you're ready to concede backup is not practical and want to get back to using the phone should everything else above fail you.