Not that I think you specifically have this misconception, but something to make clear is that the phrase "Allahu akbar" has absolutely nothing to do with battle or soldiers or anything like that. Although often translated as "God is great" or "God is the greatest," the closest translation would be "God is greater." Not just historically but even today, it's used during prayer, during extreme happiness, to show gratitude, and during times of distress and pain.

Although I can't pin point an exact first time it was said in battle, I would be shocked if it did not predate Islam. Allah was the Arabic term for God and they most likely used the phrase "Allahu akbar" in battle (along with praising a veritable pantheon of sub-deities). Off the top of my head, Ibn Kathir in his Seerah an-Nabawiyyah mentions that the Muslims responded to Abu Sufyan's comment of Hubal (a Makkan deity) being high on the day of Uhud with the phrase "Allahu a'la wa akbar" meaning "God is higher and greater." So definitely no later than that but again, I'd be very surprised if it wasn't used before then in battle. It was a very common Arabic phrase before Islam.