First, we should understand that in the early church, there were no situations where people accepted Christ without being baptized. Indeed all who came to Christ were baptized.
The closest example we might find in the scripture of a believer who was not immersed is Apollos (Acts 18:24-26). Apollos was a very spiritual man who had great fervor for his relationship with Jesus. But he had not been taught correctly concerning baptism. He knew only of John’s baptism, which was a baptism of repentance and preparation for the coming Messiah. He was unfamiliar with Christian baptism as the Apostles and early church leaders taught it. When Priscilla and Aquila instructed him, scripture says he was willing to change his ideas and submit to Christian Baptism… Baptism into Christ. They did not judge him; they merely taught him. That is our position as well.