When will the reign of the four beasts in Daniel end?
The reign of the four beasts (lion, bear, leopard, terrible beast) will end with the Second Coming of Jesus Christ. Each was absorbed by the empire that conquered it so aspects of the first three empires live within the last Roman Empire. This is what is meant by the phrase: "And as for the rest of the beasts, their dominion was taken away" (Daniel 7:12a).
The phrase, "yet their lived were prolonged for a season for a season and a time" (Daniel 7: 12b) is most probably a reference to the 45 days after the return of Jesus in which He judges the nations (Matthew 25: 31-46). All those who survive the Tribulation will be judged. The wicked will be found guilty and thrown into the Lake of Fire (Matthew 25:41, 26), while the righteous will enter the Millennial Kingdom in mortal bodies (Matthew 25:46).