According to NARA's online research guide Naturalization Records, in the United States:
From 1790 to 1940, children under the age of 21 automatically became naturalized citizens upon the naturalization of their father.
To my understanding, e.g., in the 1930s, a petition for naturalization would serve as the name change documentation for the father, since the typed form links the father's name on the ship manifest to the Anglicized name he has since adopted (unlikely to have been changed at Ellis Island).
This petition for naturalization lists the names and birth dates of the father's children, ostensibly as part of their derivative naturalization through their father. But the petition only shows the children's Anglicized names, and thus doesn't document the children's name changes from the ship manifest.
What, then, for legal chain of documentation purposes would serve as proof of the child's name change (e.g., that the child with a foreign birth certificate is the same person that was naturalized by a different name in the United States)?