The timeline of the exile is always confusing to me. The story is told by Jeremiah from Jerusalem and by Ezekiel from Babylon, each with slightly different perspectives and information, but I think it goes like this:
722 Assyria takes (Northern) Israel
609 JehoiaKIM installed as king by Pharaoh Necho
Jerusalem is a vassal state of Egypt
605 Jehoiakim rebels against Necho (stops paying tribute).
Jeremiah prophesies 70 yrs of captivity
Nebuchadnezzar claims Jerusalem and takes some Temple treasure. (Babylon and Egypt are at war)
597 Babylon's 1st siege of Jerusalem. Takes the rest of the temple treasure
18 yr old JehoiaCHIN installed as King (for 3 months)
Jerusalem is a vassal state of Babylon
1st exile relocates about 7000 of the prominent Jewish people including Ezekiel and probably Daniel
Zedekiah (21) installed as king
??? Zedekiah rebels against Babylon (stops paying tribute) and calls to Egypt for help
588 Babylon returns to Jerusalem for second (and final) siege
586 City and Temple are left in ruins
Remaining Jewish survivors are relocated