Ask a retired patrolman. That's your best bet. All else fails, most police uniforms follow the pattern of military uniforms, especially in the '30s - '50s. Ike jackets were VERY popular after WWII, particularly on motorcycle cops.
One detail that most police uniforms have is the bellowed chest pockets. British uniforms have these, so if you wanted to fake it, a tropical tan British commonwealth battle dress blouse and matching pants would work. Swap the buttons, and you're pretty close. Commonwealth uniforms didn't change much from WWII on up to the '80s, so you've got plenty to work with. Dark brown or black are options for winter wear, if you're handy with a bottle of dye.
There's a '30s vintage leather police overcoat with a mouton collar here in the shop. I've had my eye on it, but it's more than I can shell out right now. It does need some reapair, as well.