Greetings niamhiv,
Welcome to the Apple Support Communities!
I understand that you would like to update your iPhone from iOS 8.3 to iOS 8.4.1, but receive a message that your iPhone needs to finish restoring from the iCloud backup first. Even though you see the content on your iPhone that you expect, the restore may still be in progress. The attached article below explains how to check the progress of the transfer.
If you need help restoring from an iCloud backup - Apple Support
Restoring from a backup might take a while because of the size of the backup or the speed of the Wi-Fi network. For example, iCloud can take approximately 1 to 4 hours per gigabyte during a restore process.
There are several steps you can take:
- In iOS 8 or later, check the progress of the transfer on your device under Settings > iCloud > Backup. In iOS 7, go to Settings > iCloud > Storage & Backup. If the restore process is underway, the iCloud Backup setting is dimmed.
- Make sure that your device is connected to the Internet with reliable Wi-Fi.
- Make sure that your device is plugged in to a power outlet.
- Check the Apple progress bar that appears while transferring a backup.
Cheers