No, because an iCloud backup is specific to an Apple ID. I am bit confused by you saying they have phones but they also have new phones? If you choose to change An Apple ID on a device if you just change the Apple ID then when it comes to iCloud services the phone will ask you if you want to keep items on the phone from the old Apple ID so they can then be merged with iCloud for the new ID. You may find this option keeps a lot of things. Do all this setting up on the old devices so when you transfer to new devices you aren't messing with too many changes at once.
Help your child set up an iPhone, iPad, or iPod touch --> https://support.apple.com/HT205763
Click on this link --> https://support.apple.com/HT201269 to learn how to "Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch". Select one of the three options: Quickstart, iCloud, or iTunes/Finder.