Assyria and ancient Babylonia (not the Babylonia of Nebuchadnezzar) had many of the same Gods. The religion is labeled the Assyro-Babylonian religion. This is where the legend of the sea monster Tiamat (a name often given to dragons) comes from.
The religion resulting from these nations were the first to have gods with wings. Assyria popped up roughly the same time as Sumeria about 4000 years ago. Babylonia popped up with Hammurabi developing the first real written law about 1700 BC. The religion that these two nations shared become THE religion of the time.
For some reason (I'm still looking into this) the Babylonians and Assyrians saw fit to put wings on their gods. It certainly made them more threatening and given that both civilizations were of warrior tradition, a god with wings would certainly be a good fighter.
With the amount of influence that these nations had at the time, it's not surprising that the Jews would give wings to God's army of drones, the angels. At the time it was not uncommon for people to believe that their god(s) was/were doing battle with the mythology of other nations when they went to war. The Assyrians and Babylonians flying gods would certainly be foreboding to the imagination.
As a side-note, the above god is Nisroch.
Nowadays, we can hold the concept in our minds that a person without wings can fly (see Superman) but at this period of time, it didn't make as much sense.
Now the idea of angels sprouting wings from the Babylonians is a theory. But, it's very apparent that the Babylonians and Assyrians did it first. That being said, I repeat what I said earlier about stories of angels simply being tall humanoids with no wings.
People will believe what they want, and the evidence in this case is circumstantial as far as I know. But it is very interesting.