Someone here may have an answer, but the factory was turning out several hundred handguns a day, seven days a week, and they were in high demand, so my guess is they were shipped out as fast as they were made and logistics allowed.
The numbers per month, just for the U.S. Army contract varied from 996 to 12,236. Then there was the Navy contract, OSS and DSC. 38-200 monthly procurement numbers, from all contracts were between 880 to 19,616. Those are just the S&W numbers and do not include various Colt, High Standard, H&R, etc. handguns.
The OP specifically mentioned US Navy revolvers. The one that I had (sold a few years ago) was in a shipment of 1000 that shipped direct to the Navy, under the Navy Contract on December 17, 1942.
It is my understanding that shortly thereafter, the Army took over all shipments and there were no more Navy direct contracts.
When I got my Factory Letter (copy attached) I also received a copy of the original invoice to the Navy (also copy attached) so you can see the details of the contract numbers.