This is a problem that I have with most Apostolic churches today. They are "formula" driven. Jesus name baptism isn't a "formula" or a "liturgy" to be repeated over a person by an ordained "priesthood". Let's look at Paul's baptism closely,
(Acts 22:16 KJV)
(16) And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord.
Paul was told to be water baptized, and wash away his sins, calling upon the name of the Lord. You see, Ananias didn’t repeat a formula over Paul. At baptism Paul himself called upon the name of Jesus for the remission (forgiveness) of his sins. Here’s the point, it doesn’t matter what a MAN speaks over you at the moment of baptism. What matters is that YOU are calling on Jesus to wash away your sins. If YOU don’t call on the name of the Lord for cleansing, you will not receive remission. You could have a Catholic priest baptize you in the names of Larry, Moe, and Curly, but if YOU are calling on the name of Jesus, your sins will be remitted. Or, you could have a Pentecostal preacher baptize you screaming, “In the name of Jesus!”, 100 times…but if YOUR not calling on the name of Jesus, the baptism is ineffectual.
Now, about the hypothetical man form the 16th century. If his heart was calling on the name of Jesus at his baptism, it doesn’t matter what “formula” or “liturgy” was used…the man’s sins were washed away. If he was baptized as an infant and chooses to be re-baptized, it doesn’t matter what’s said over him if his heart is calling on the name of the Lord.
God is concerned with the disposition of the heart…not the wording of a “formula” or “liturgy”. God's grace can cut through the muttered traditions of a priest to forgive the sins of a sincere heart.
If you look very closely at the Apostolic church of the NT... they didn't have a "formula". It was simply about the repentant calling on the name of Jesus while being burried with him in baptism.