I'm uncertain if you can back this up with the iTunes back-up, but I know you can copy the world folder and paste it back afterwards.
Step 1 : Press the "Recents" button on your device and swipe Minecraft to close it. This will completely close the game and stop it from running in the background.
Step 2 : Download a File Manager app on your device, or connect it to computer. This will allow you to access the file system of your Android device.
There are several free file explorers available on the Play Store. ES File Explorer is one of the most popular.
If you connect your device via computer, you can navigate through the folders just like you would with Windows folders.
Step 3 : Start the File Manager app on your device. You will see a list of folders on your device. You may have to open the "sdcard" folder to see all of your system folders, depending on your device.
You don't need a file manager app if you connect your device to the computer.
Step 4 : Open the games → .com.mojang → minecraftWorlds folder. If you are using your computer, connect your device and select it from the Computer window (Windows) or your desktop (OS X).
This will display a list of folders labeled with all of your created worlds, both Creative and Survival.
Step 5 : Open the folder for the Creative world you want to transfer. You should see a "level.dat" file.
Step 6 : Copy the "level.dat" file.
Android - Press and hold the "level.dat" file and select "Copy" from the menu that appears.
Computer - Right-click on it and select "Copy".
Step 7 : After the backup you go to the same folder and paste your copied files there. If your daughter(s) boots up minecraft again she should have her worlds again.
NOTE : This only saves the world, the player data might not be saved.
For iOS, as Pansy said in the comments. iOS devices dont allow you to access their files so you can't do the above steps unless you pay for software that allows you to access those files.
Copy your worlds with PhoneView
PhoneView ($29.95 Mac) from ecamm, the makers of Printopia, is an
application that can access your iOS devices filesystem in order to
access the information stored on your device. This includes the
documents that are created by each app on your device. It just so
happens that for Minecraft PE, the world files are stored in a manner
that is easy to back up and restore. You do not need to jailbreak your
device in order to use PhoneView to back up and restore your Minecraft
Source for IOS : HERE