Harry and Meghan's children do have a certain amount of protection as the Celebrity Children's Act passed in 2013 (SB 606)
It was passed at the urging of celebrities like Halle Berry and Jennifer Garner - it strengthened anti-paparazzi laws by increasing penalties for harassing children under 16, making it a misdemeanor to intentionally alarm, annoy, torment, or terrorize them due to their parents' public status, adding jail time (up to a year) and fines (up to $10,000), and allowing parents to sue for damages and profits, protecting kids from becoming tabloid fodder.
The law aimed to stop paparazzi from pursuing celebrity children aggressively in public, treating them as "tabloid fodder," and giving families greater protection in their daily lives, like going to school.
Under the law Harassment includes actions taken while attempting to record a child's image or voice without consent, such as: Following the child or parent. Lying in wait. Or other conduct that seriously alarms, annoys, torments, or terrorizes the child.
It also introduced the requirement for parental consent for publications to use pictures of celebrity children. So when you see paps taking pictures of celebrity kids - such as Archie a Lilibet photographed trick-or-treating at Halloween you know that Harry and/or Meghan has consented.