NELSON - who are the names of the 3 nephites written in the book of mormon....and were are the 3 nephites right now?

JOEL - When Jesus appeared to the Nephites He chose twelve disciples to help Him minister to the people:

"And it came to pass that on the morrow, when the multitude was gathered together, behold, Nephi and his abrother whom he had raised from the dead, whose name was Timothy, and also his son, whose name was Jonas, and also Mathoni, and Mathonihah, his brother, and Kumen, and Kumenonhi, and Jeremiah, and Shemnon, and Jonas, and Zedekiah, and Isaiah—now these were the names of the disciples whom Jesus had chosen—and it came to pass that they went forth and stood in the midst of the multitude." (3 Ne. 19: 4)

Here is what Jesus said to the three who asked to linger on the earth:

"And he said unto them: Behold, I know your thoughts, and ye have desired the thing which John, my beloved, who was with me in my ministry, before that I was lifted up by the Jews, desired of me.
Therefore, more blessed are ye, for ye shall never taste of death; but ye shall live to behold all the doings of the Father unto the children of men, even until all things shall be fulfilled according to the will of the Father, when I shall come in my glory with the powers of heaven.
And ye shall never endure the pains of death; but when I shall come in my glory ye shall be changed in the twinkling of an eye from mortality to immortality; and then shall ye be blessed in the kingdom of my Father.
And again, ye shall not have pain while ye shall dwell in the flesh, neither sorrow save it be for the sins of the world; and all this will I do because of the thing which ye have desired of me, for ye have desired that ye might bring the souls of men unto me, while the world shall stand." (3 Nephi 28:6-9)

Here is what Mormon said of them:

"Behold, I was about to write the names of those who were never to taste of death, but the Lord forbade; therefore I write them not, for they are hid from the world.
But behold, I have seen them, and they have ministered unto me.
And behold they will be among the Gentiles, and the Gentiles shall know them not.
They will also be among the Jews, and the Jews shall know them not.
And it shall come to pass, when the Lord seeth fit in his wisdom that they shall minister unto all the scattered tribes of Israel, and unto all nations, kindreds, tongues and people, and shall bring out of them unto Jesus many souls, that their desire may be fulfilled, and also because of the convincing power of God which is in them.
And they are as the aangels of God, and if they shall pray unto the Father in the name of Jesus they can show themselves unto whatsoever man it seemeth them good.
Therefore, that they might not taste of death there was a change wrought upon their bodies, that they might not suffer pain nor sorrow save it were for the sins of the world.
Now this change was not equal to that which shall take place at the last day; but there was a change wrought upon them, insomuch that Satan could have no power over them, that he could not tempt them; and they were sanctified in the flesh, that they were holy, and that the powers of the earth could not hold them.
And in this state they were to remain until the judgment day of Christ; and at that day they were to receive a greater change, and to be received into the kingdom of the Father to go no more out, but to dwell with God eternally in the heavens." (3 Nephi 28: 25-30, 38-40)

So we don't know which of the three disciples remained on the earth but their names are among those given in the 3 Ne. 19: 4 scripture quoted earlier. We also don't know exactly where they are; we only know that their bodies were changed or translated, so they would never die nor be tempted of Satan. At the second coming of Christ they will be changed in the twinkling of an eye to a glorified resurrrected state. They were given a charge to minister to all the children of Israel and the Gentiles on this earth, but noone will know who they are. The primary responsibility of the Three Nephites is missionary work, the gathering of Israel into the true church and fold of God.
Both Mormon and Moroni said they appeared to them and others three hundred years after the coming of Christ to the Nephites (3 Nephi 28: 26, Mormon 8: 10-11, 4 Nep. 1:30-37), and there have been reports of them appearing to many other people since then. However since the scriptures tell us that noone will know who they are (3 Nephi 28: 27-28), claims of their appearances to people may be somewhat speculative.