"It has what is called a pencil barrel."
No, it has a standard or tapered barrel. "Pencil barrel" is incorrect terminology and we really don't need to promote its usage.
But other than knowing the correct terminology to describe the barrel I am confused on this gun. You say 4" barrel but it looks like 5" to me. Also, you state the caliber is "38 S&W" (word Special not present?) and that would not be correct for a Military & Police Model intended for the U.S. market. It is the pre-war "long action" design but I wonder if it's not actually a WW II Victory Model made for Great Britain and subsequently nickel plated? The serial you provided doesn't fit anything the gun appears to be.
The target stocks are post 1968 and not original. Is there a hole in the bottom of the butt? That would confirm it's a military contract model.
BTW- Just to add to the confusion, the Detroit PD is known for using nickel plated 5" revolvers. On top of the other possibilities it might be an issue DPD gun. But I think they were stamped on the back of the grip frame.
I keep coming back to the caliber. Screams Victory Model for England during WW II.
This is what a British Victory Model would look like in standard configuration. Note the lanyard ring on the butt. That's why I asked if there's a hole where the ring used to be.