They didn't exactly start at 15,000.
The earlier Remington-Beals Army and Model 1861 (Old Army) that preceded the 1863 (New Army) were all numbered in the same serial number series, and production of each improved version was phased in as production continued and older parts were used up. But sometime in 1863 production on the New Army within the 15,000’s (something) started. Because the Civil war was on Remington was turning out revolvers as fast as possible. During the war, the Union purchased some 114,713 Army revolvers of all styles from the company. Next to Colt they were the second largest supplier of Cavalry revolvers to the U.S. Army.