In an early episode of I Dream of Jeannie, the main characters are transported from 1960s Florida to Baghdad "2000 years ago". The episode was filmed in 1965, so this puts them in 35 BCE. Many of the ancient characters reference Allah in one way or another.
Two questions: does Baghdad exist? And would anyone have worshipped Allah in Mesopotamia at that time?
I don't know much about your first question regarding whether Baghdad existed, other than to say that the name Baghdad is pre Islamic, and the Babylonian city of Baghdadu did exist and refers to the same geographical region as Baghdad does today.
As for whether anyone in Mesopotamia worshipped Allah, it depends how you conceive of this.
One must consider that the name Allah to refer to the one, unitary, unbirthed and eternal monotheistic God of Islam and Abrahamic faiths wasn't used or perhaps didn't exist until Muhammad's revelations. The name or word Allah itself is most likely an amalgam or contraction of "al ilaah" which literally means "the God". The generic word for god in the Arabic of the quran is also ilaah, and it is only when referring to "The God" that the name Allah is used. So we can say that no one would have used the word Allah to refer to a God that they worshipped.
But if Allah is unbirthed and eternal, then that means Allah as conceived by Muslims (and by extension this is also the way God is conceived of by Christians and Jews, generally speaking) existed at this time. Jews in Iraq as monotheists would have worshipped "the God", which Muslims and many non Muslim but Arabic speaking monotheists would call Allah. So from this perspective we can say that they worshipped the monotheistic God of Abraham, who is Allah (among other names he has been called in history), but they would not have called him Allah.