In 586 BC Nebuchadnezzar conquered Jerusalem, and a great deal of the population was brought to Babylon. What they thought of that we can read in psalm 137, e.g. vs 8 en 9:
O daughter of Babylon, who art to be destroyed; happy shall he be, that rewardeth thee as thou hast served us.
Happy shall he be, that taketh and dasheth thy little ones against the stones.
Ezekiel is the prophet of the Exile. He had been carried of to Babylon at an earlier time, and as long Jerusalem is not conquered he tries to convert the people. After it has been conquered he comforts the people. He teaches that Gods presence is not limited to the country of Israël. It is true, sacrifices cannot be brought in the Temple, but God can be worshipped just as well.
Lamentations describes the destroyed Jerusalem, and especially the grief its destruction causes. The well-known stories of the book of Daniel also occur in that time: the three friends in the fiery furnace, the dream of Nebuchadnezzar and his madness, and the story of the writing on the wall, and the lions.
In these days these stories are looked upon more critically. One of the reasons for that is the accuracy of the predictions of the second part. Therefore it is thought they are written afterwards.
(And many people also have problems with the fiery furnace).
The prophecies end in the time of the Maccabees, about 166 BC, and therefore the book Daniel should have been written at that time.
The opera Nabucco, with its Hebrews' Chorus, occurs in that time. Nabucco is Nebuchadnezzar.
Not all Jews were miserably in Babylon, some were quite succesful.
Inscription of Nebuchadnezzar (Ishtar Gate, Pergamum museum Berlin)
Only a part of the inhabitants of Judea were deported. As described in Jeremiah 40 – 43 Gedaliah became chief of the people that stayed, but he was killed and the murderers fled to Egypt, and took Jeremiah with them, against his will. In Egypt lived quite a few Jews at that time. Some as mercenaries. Jeremiah prophesies against them in Jeremiah 44. He says that all the disaster came upon them because they have left God, but they say it is because king Josiah has abandoned the service to the Queen of the Heavens ( Astarte).
They had built a temple for God on the island of Elephantine in the Nile, a competitor of the temple of Jerusalem. Maybe they were not that much heathens.
Some think that these Jews, when they fled during the reign of the evil king Manasseh to Egypt, they took the Ark of the Convenant with them. In the description of the fall of Jerusalem the Ark is not mentioned, neither is it mentioned in the stories about the stay in Babylon.
From Egypt it is thought to be brought to Ethiopia, and it is still there, in the church of the Holy Mary of Zion, in Axum.
In 539 BC Babylon was conquered by Cyrus, and the Exile ended. It lasted 47 years. Cyrus did not destroy Babylon, contrary to psalm 137.