I got hit with the Slide to Upgrade Bug when upgrading my iPad from iOS 7.1.2 to iOS 9 this weekend. Apple’s solution at https://support.apple.com/en-us/HT205239 is to force restart your device, connect to iTunes, take a backup if you need and restore your device from the backup. For me, taking the backup failed with an error “You iPad has not be setup”. If you are able to sync, you can sync your photos with the Photos application. If even that did not work, there is still hope.
The way I recovered my photos from my iOS device stuck on Slide to Update screen was really simple. I plugged my device into a computer without iTunes. This mounts the photos folder on the computer and allows you to copy all the photos to a safe place. I used Ubuntu 14.04 for this, but if you have Windows, that will do just fine. Remember that it should not have iTunes.
Since, I did not have a spare computer I used the free Virtual Box to create a Ubuntu 14.04 virtual machine. Connected my iPad to the computer. Then used the Device -> USB menu in Virtual Box to connect my iPad directly to the Ubuntu Virtual Machine. Force reset the iPad. Since this was the first time I connected my device to the Virtual Machine. I had to unlock the iPad and click on Trust button in the dialog on my iPad to allow the computer access to the iPad. The iPad was then mounted and I could simply copy my photos off the iPad onto a hard disk.
If you are stuck on the Slide to Update screen and nothing happens when you slide, repeat sliding very quickly. It took me about 10 very quick slides before it gave me the unlock screen. Once I entered my pin, I got the dialog to Trust the computer.
Along with photos, you can still recover some more data from your device. Some apps have file sharing enabled and the shared data from these apps can be copied even if the backup or sync is failing in iTunes. To do so, you will need to connect your device to iTunes and click on the apps tab. At the bottom of the page there is a section labeled File Sharing. You can select the app from that list and save all the files shown on the right side. The details on how to use this feature from Apple at https://support.apple.com/en-us/HT201301.
I also managed to check the syslog and get a crash dump. It seems that the reason for the failed Slide to Update screen is due to the data migration process (com.apple.datamigrator) which keeps crashing while migrating data, but it is pure speculation on my part.
Feel free to drop me a comment if you find this useful or need more instructions on how to set up the Virtual Machine.