No one has any rights but parents can release them for repatriation to their original country or the kids could just leave when they are old enough.
The child doesn't have an original country. There is no record of them being born or who their parents are because there is no government. There is no communication with the outside so records can't be held externally.
To be clear I think you're describing a horrific dystopia that should be opposed by any ethical human being.