Assyria conquered Israel, then Babylon conquered Assyria and Judah, then Persia conquered Babylon, then the Seuclid Empire conquered Judah, then the Judeans revolted, then Rome conquered Judah, then the Islamic Caliphate conquered the Byzantine Empire (the remains of the Roman Empire). The Ottoman Empire conquered Judah from the Cusaders who had conquered it from its Islamic rulers. Then the British Empire took it from the Ottomans.
The Assyrian Empire conquered the northern kingdom of Israel in 722 BCE, renaming it as the Assyrian province of Samaria.
The Babylonian Empire.
the capital city of Judah is Jerusalem. both were conquered by both Israel and Judah. Jerusalem.
Roman.
The Neo-Babylonian Empire (also called the Chaldean Empire) led by King Nebuchadrezzar conquered the southern Kingdom of Judah in 586 BCE.
rome
The wording of this question is confusing. "Judah" was the name of the country and after Judah was conquered by the Assyrians, it never asserted independence again under that name. "Israel" is also confusing because Israel was the kingdom to the North of Judah whose inhabitants stopped associating themselves with that State by the time of the Persian Empire. If your question means to say "When were the Judeans allowed to return to Canaan following the Babylonian Exile?" the answer is the 520s-510s BCE.
Judah sought an alliance with Assyria against Israel, which was threatening to invade Jerusalem and depose the king. Assyria conquered Israel in 722 BCE and made the former kingdom into the province of Samaria. Assyria then turned its attention to Judah, with the intention of extending its empire southwards, but found the fortified city of Jerusalem too hard to defeat.
How: By military invasion and conquest. The Assyrians overthrew Israel; the Babylonians conquered Judah. When: Israel, 722 BC; Judah, in three invasions between 605 and 586 BC. Why: As punishment for all manner of disobedience to God.
There was no Israel in 587 BCE. Israel was destroyed by the Assyrians in 722 BCE.If perhaps the question was referring to the Kingdom of Judah which would be destroyed one year later by the Babylonians, the King of Judah at that time was King Zedekiah son of Josiah.
Britain from 1919-1948.
The kingdom of Israel was conquered by the Assyrians in about 720 BC. when it ceased to exist. However, if by "Israel" you mean the territory that composes present day Israel, it was variously named. The southern kingdom of Judah became the Roman territory of Judea which was ruled as a part of the province of Syria. Hadrian renamed the area Palestina Syria. The Ottoman empire ruled after the fall of the Roman empire and the British had a ruling interest in the area up until the establishment of the modern country of Israel.