You backed up the iPhone to where? iCloud? PC? Mac? How was the back up performed? Is the new device up to date? Sometimes out of the box iOS will be a couple minor updates behind. Good internet connection? Adequate space for the back up you made?
If its iCloud you will need to be logged into the same AppleID/iCloud account. Goto Settings > Apple ID (your wife's name) > iCloud > Manage Storage > Backups
From there you should see that accounts device back ups. Do you see anything in there? If not, again check for iOS updates, log out and back into iCloud (verify its the right username), reboot the iOS device.
The back ups contents are device based app data for 3rd party apps that don't utilize iCloud cloud save/share natively. So while it varies between individual its not typically the end of the world to restore as new. This will all depend on how you used iCloud prior, if everything was turned on and there was space in iCloud then the essentials will come back with the AppleID log in (contacts, photos, messages, calendars, settings, notes, etc etc etc....
I never restore from a back up with a new device, just logging back into my AppleID and redownload my apps (gives me the opportunity to tidy up my app install base).
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With Mac (and probably PC) you will need to be logged into the the same user account on the Mac/PC the backup was made. You should be able to find it if you open the device in Finder (iTunes for PC?) go to the device section where back up is and click 'manage backups'.
Again though, as long as you didn't have iCloud settings all turned off a large chunk of the data will re-populate on its own. You just might need to redownload and set an app back up the way you liked it, like a 3rd party camera app settings. Or re-log back into a lot of your apps that are web based anyway.
Good luck!