The USCIS has no authority to change anybody's name. The request for a name change on the N-400 form is a request to a court that administers your Oath of Allegiance to change your name at the same time, such that if the court grants your request your new name appears on your Certificate of Naturalization.

You appear to be talking about changing only their first names, not their surnames? My approach would be to play it safe by doing that before filing the N-400/N-600's, and documenting the name changes with your naturalization petition.