You lose them.
Well, in a chapter 7 you lose them. In a chapter 13, you have to pay their value to your creditors as part of your chapter 13 plan.
I know, you want to argue that it is their money which they have worked for and saved up. That doesn’t matter. Anything your minor children “own” is actually your property. Otherwise, you could transfer the title of your home to your 3 year old and effectively protect it from your creditors, since it’s his house now, and not yours. This may seem like a harsh example, but it is the bankruptcy trustee’s job to look for unprotected (nonexempt) assets which he can seize and use to pay your creditors. Anything of value you’ve given to your children (like cash in a savings account) is something that you may lose in the bankruptcy.
That being said, so long as the accounts are small, the trustee will probably consider them de minimus and leave them alone.
You may want to argue that grandma and grandpa gave them that money for college. If they’re a minor, then it’s still your money. It may be protected if the money has been slowly placed in a 529 college savings plan, but this is not the post to discuss that in detail.
The short answer is that anything your children own is YOUR property, and you had better discuss it with your attorney to determine whether or not it is protected.