If the world is to end what will happen to humanity, nevertheless God?
Addressing the last part of your question first, the end of the world would have no effect on the existence of God. God existed before He created all that we know (Genesis 1), and there is no reason to believe that His choice to destroy the world would end His existence. An example might help to show this. When a potter sits down at his wheel he creates a pot. It sits on the shelf for a while, and then he decides for some reason to shatter it. He lived before he made it, and he continues to live after he destroys it. So it is with God. "And, Thou, Lord, in the beginning hast laid the foundation of the earth; and the heavens are the works of thine hands: They shall perish; but thou remainest; and they all shall wax old as doth a garment; And as a vesture shalt thou fold them up, and they shall be changed: but thou art the same, and thy years shall not fail." (Hebrews 1:10-12)
More important, perhaps, is what will happen to humanity when the world is destroyed. There are scriptures that address just that question.
"But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up. Seeing then that all these things shall be dissolved, what manner of persons ought ye to be in all holy conversation and godliness, Looking for and hasting unto the coming of the day of God, wherein the heavens being on fire shall be dissolved, and the elements shall melt with fervent heat? Nevertheless we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness. Wherefore, beloved, seeing that ye look for such things, be diligent that ye may be found of him in peace, without spot, and blameless." (2 Peter 3:10-14)
Peter says that something of what we are will survive the destruction of the world and live on in "new heavens and a new earth." Some people claim that this teaches that we will live on in a physical way on a new physical earth. I think other scriptures show this to be on a spiritual level, rather than physical; that is, men's spirits will continue to live in a different form, receiving eternal joy in heaven or eternal pain in hell.
What will that new spiritual form or "body" be like. Paul addresses that in 1 Corinthians 15:35-58. I summarize it here, but look it up and read the whole passage. Paul says we will be changed into a "spiritual" and "immortal" body. He concludes by saying, "flesh and blood cannot inherit the kingdom of God; neither does corruption inherit incorruption." So mankind will be given a new, spiritual body.
Going a little beyond your immediate question, though, it is important to ask what will happen to those immortal bodies. This is answered in Matthew 25:31-46. (Again, please read the whole passage for yourself.) Jesus said there will be a judgement, a separation. Those who have believed in and obeyed him will be separated from all others. The final verse of that passage shows what will happen to those new, spiritual bodies. "And these [unrighteous] shall go away into everlasting punishment: but the righteous into life eternal." Righteousness, however, is only accounted to those who avail themselves of the grace of God through the sacrificial death of His son, Jesus the Messiah. Without his blood covering our sins none of us would be counted as righteous.