Hi Mike,
So, an iCloud backup is a backup of all of the data on your device only. If you do not have iCloud Photos turned on for your device (i.e., you do not currently have all of the photos showing on your device), then only the photos showing in your Camera Roll will be backed up.
You can set your device up again using either of the methods to set it up again by erasing it first. Be sure you do not turn on iCloud Photos (if you and your wife are using the same iCloud ID - if her photos are under her own iCloud ID, then there would be no issue at all because photos in iCloud Photos only download to devices that are signed into that iCloud account and that have iCloud Photos turned on).
But the Quick Start is probably the easiest method to use if you are able to use it.
If you cannot, then you can make an iCloud backup of your old device. Make sure that under Settings>your name>iCloud>Photos, you do not have iCloud Photos turned on before you back it up. If that option is turned on, then turn it off before making the backup.
Here are the support articles for erasing your device:
How to erase your iOS device - Apple Support
Once you see the "Hello" screen, follow either the "Quick Start" or for the "Use iCloud" to set up your device again.
Best,
GB