ANSWER: In a very real sense, we all nailed Jesus to the cross as he died not just for all sins collectively but also the ones you and I commit. Christ is just as much a personal Savior as he is one for the whole world. That said, your question concerns who, in the first century A.D., literally nailed Christ's flesh into wood in order to suspend him above the ground as a warning to fear the punishment of Rome.
The only thing we know for sure in regard to who nailed him is that one or more Roman soldiers did the real work of affixing Jesus to the cross (actually, in was a stake).
In the first century the Romans occupied Judea and had an official Roman appointed king over the people (Herod the Great was the first appointed king from 37 to 4 B.C.). This means that the Jews in Judea did not have the authority to carry out the death penalty for capital crimes. If they had possessed such power, they no doubt would have nailed God's son sooner without having to manipulate the hated Romans to do it.
Then Pilate said to them (Jewish religious leaders who wanted Jesus dead), "You take Him and judge Him according to your own law." But the Jews said to him, "It is not lawful for us to put anyone to death" (John 18:31 - 32, HBFV).
The empire, through the Roman Prefect Pontius Pilate (who served in this capacity from 26 to 36 A.D.), ultimately made sure the sentence against Jesus would be carried out completely by having guards around the crosses of those who were condemned (Matthew 27:54).
No record exists of the names of those Roman soldiers who nailed Jesus to the stake. As far as what happened to these people after his death, there is also no historical record. What is ultimately important is that those who nailed the Son of God were forgiven, a fact only recorded in the book of Luke (Luke 23:33 - 34).
Here are some things to consider regarding the death of Jesus at Golgotha.
The same injustice that condemned the innocent Son of Man also accomplished God's justice
The Satanically-inspired plot to murder the Lord became the means mankind could be released from the power of the devil and his deceptions
Our Savior's ignominious death through the torture of being nailed the cross became a light of hope to a world living in spiritual darkness
Since the life of Jesus was worth more than the lives of all human beings combined, his death through being nailed on the cross was more than sufficient to pay for the sins of the world. He now lives as man's constant mediator to God.