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.
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.
Israel had a border with Assyria. Judah had Israel as a buffer state to protect them from Assyria.
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.
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.
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.
In the Old Testament, only Assyria conquered Israel. This occurred in 722 BCE and resulted in many of the Israelites being deported and Israel henceforth ceased to exist.A century and a half later, Babylon conquered the smaller, southern kingdom of Judah, which is sometimes confused with Israel.
Britain from 1919-1948.