At which times in history have there been a sovereign mainly Jewish state controlling Jerusalem? The only time in history after the Babylonian exile that I can recall is during the Hasmonean dynasty. But have they been sovereign during other times as well?
Judaism grew out of the Canaanite religion, in the Canaanite area. So for most of the early existence of the region it was split up in independent areas, probably some sort of city-states. In the later period two confederacies seem to have emerged, one centered around Kadesh, the other around Megiddo.
The Canaanites lost independence some time during the 2nd millenium BC, maybe around the middle of the century, and after that it was frequently fought over by Egypt, Assyrians, Hittites etc.
Classical Hebrew as a language showed up around a 1000 BC, at the same time the so called "United Kingdom" is most often thought to have happened. According to the Bible first King David (of Goliath fame) and then King Solomon are supposed to have ruled a united Kingdom of Israel. As we see from the story of David and Goliath, these kings have become mythical, and it's hard to separate myth from fact, and there are scholars who say that they don't think there ever was a united kingdom. But nevertheless there was a kingdom of Israel (often called Kingdom of Samaria to lessen confusion with the united Kingdom of Israel) and a Kingdom of Judah, both practicing similar proto-Jewish religions and speaking Hebrew, and as such they can at least be called Hebrews, if not actually Jewish.
This period of independence seems to have lasted from around 1000 BC to 732 BC, when Assyria invaded.
Judaism as we recognize it today, with monotheism and the first books of the Bible (although some of the sources are much older), only arrived during the Babylonian exile. And with that definition you are correct; Only with modern Israel and the Hasmonean dynasty has there been a sovereign Jewish state in the Israel/Judah area.
The Hasmonean dynasty ruled independently between 110 BC, when the Seleucid empire collapsed, and 63 BC when the Romans conquered the area.
Therefore we can list the following sovereign times for Jerusalem and surrounding "Israelite" lands:
Which ones of these you call "Jewish" is largely a matter of opinion (the development of Canaanite religion into Judaism is gradual), but the Judaism that we today recognize developed mostly during and after the Babylonian exile in the 6th century BC.