What happens at the last trump?
At the last trump, the seventh angel sounds and the mystery of God is finished: the dead in Christ are raised incorruptible, the living are changed, and the rapture of the Church takes place as the kingdoms of this world become the kingdoms of the Lord and His wrath begins.
The last trump spoken of in 1 Corinthians 15:52 is not a symbolic voice but a literal trumpet, and it is identified with the seventh and final trumpet of Revelation. "In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed." This resurrection and change of the living saints is the rapture, and it does not happen early in Revelation but at the sounding of the seventh trumpet.
Revelation 10:7 ties this moment directly to that seventh trumpet: "But in the days of the voice of the seventh angel, when he shall begin to sound, the mystery of God should be finished, as he hath declared to his servants the prophets." The mystery of God is described as Christ in you, the hope of glory, and this mystery is brought to completion, fully revealed and glorified in the Body of Christ, only when this trumpet sounds. Until then, the Church remains on earth, still waiting.
When the seventh angel actually sounds in Revelation 11:15-19, great voices in heaven declare, "The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever." The rule of Antichrist and Satan begins its transition to the reign of Christ. The twenty four elders worship, declaring that God has taken to Himself great power and has reigned, that the nations were angry, and that His wrath is come. At this same moment comes "the time of the dead, that they should be judged," and reward is given to the prophets, the saints, and all who fear His name, small and great, while those who destroy the earth are destroyed. This is described as the first resurrection, and all of it is determined at the sounding of the last trump.
Also at this time the temple of God is opened in heaven and the ark of the testament is seen there, showing that it has already been removed from the earth and is in heaven before the Church itself arrives there. So at the last trump, several things happen together: the dead in Christ are raised, the living are changed and gathered to the Lord, the mystery of God concerning the Church is finished, the wrath of God begins to be poured out on the world, the dead are judged and rewarded, and the kingdom begins its open transition from the Antichrist's rule to the eternal reign of Christ.
The scriptures
In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.
But in the days of the voice of the seventh angel, when he shall begin to sound, the mystery of God should be finished, as he hath declared to his servants the prophets.
And the seventh angel sounded; and there were great voices in heaven, saying, The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever. And the four and twenty elders, which sat before God on their seats, fell upon their faces, and worshipped God, Saying, We give thee thanks, O LORD God Almighty, which art, and wast, and art to come; because thou hast taken to thee thy great power, and hast reigned. And the nations were angry, and thy wrath is come, and the time of the dead, that they should be judged, and that thou shouldest give reward unto thy servants the prophets, and to the saints, and them that fear thy name, small and great; and shouldest destroy them which destroy the earth. And the temple of God was opened in heaven, and there was seen in his temple the ark of his testament: and there were lightnings, and voices, and thunderings, and an earthquake, and great hail.
Even the mystery which hath been hid from ages and from generations, but now is made manifest to his saints: To whom God would make known what is the riches of the glory of this mystery among the Gentiles; which is Christ in you, the hope of glory: Whom we preach, warning every man, and teaching every man in all wisdom; that we may present every man perfect in Christ Jesus:
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.