A case was brought to Rav Moshe Feinstein in 1960 of a boy whose mother had converted Reform and then given birth to him, at which time he had a brit mila. Now, as the boy was approaching his bar mitzvah, the family had become more religious and the mother realized that her son needed to convert, since her conversion was deemed invalid. She agreed to have her son go to the mikveh, but did not want him to have a "second" brit mila (meaning הטפת דם ברית - taking blood from the location of the brit mila), since he was a weak child.
Rav Moshe ruled that even though it would be ideal for the son to have הטפת דם ברית, nevertheless since his original brit mila was done for the purpose of a religious brit mila (and not merely a hospital circumcision), then that could count and preclude the need for any further procedure. He notes that it would be great if there were three halachically acceptable people who had witnessed the original brit mila, but even if there were not, there mere fact that his brit mila was publicized would suffice.
Rav Moshe also cautions against hiding any of the facts from the family, since eventually things will become known and it is best for them to be fully aware of the full scope of their situation.