Almost everything I know about this era of history comes from the bible and the latest episode of Hardcore History. When the Babylonians took the Jews captive, was that a big deal or was it just business as usual for them? When Cyrus set returned them to Jerusalem, were they getting special treatment because they were especially important in some way, or did he do the same with dozens of other displaced peoples who didn't cover half the globe with offshoots of their religion? When the big empires of the era made decisions, was Israel's reaction a factor?
So, this is a bit of a tough and still hotly debated issue. The presence of the Hebrews in the ancient world is attested, but the importance of a united Davidic kingdom and later Israel and Judah is not. There are some few scholars who even maintain the Davidic/Solomonic heyday of the region is little more than myth.
That being said, I think it is best to think of Israel/Judah as regional powers. When Egypt or Assyria made decisions, did they consider the result in Jerusalem or Samaria? Very doubtful, unless they had some specific interest. Did Edom or the Philistines pay attention? Almost certainly. In essence, I think the evidence points to both the united kingship and the resulting states of Israel and Judah as being fairly standard minor kingdoms in the region, like dozens of others.
Similarly, Cyrus' thought process is obviously unknowable, and the bible is our only real source for his proclamations on the people of Judah, which obviously makes it a bit tricky. But Cyrus is noted in several documents--perhaps most notably something called the Cyrus Cylinder--as repatriating captive peoples. It seems that he either made this a policy of his, or at least wanted to be seen as doing so.
Thus, my guess is, religious considerations aside, Israel and Judah were fairly unexceptional Iron Age kingdoms in the Levant, and their repatriation to Cyrus would have been just a de facto bureaucratic decision--"this is what we do with captive peoples." That's the short version from a bachelor's degree level guy!
You need to read The Bible Unearthed from Finkelstein. But in short: no, the Hebrews at this time are at most a nuisance for the big empires of Egypt / Assyria / Babylon. The moment they tried to be something more they were crushed like an ant. See Necho II killing Josiah at Megiddo and then taking over his kingdom for a very relevant example.