The .32 S&W Long is the same as the .32 Colt New Police cartridge. The "Special" is a bit of an odd moniker though.
The following cartridges are interchangeable;
.32 S&W Long
.32 Colt New Police
.32 Colt Police Positive
Note that there was a .32 Short Colt and a .32 Long Colt but these are obsolete cartridges. I'm not sure, but I think the .32 Long Colt was also called the .32 Police or .32 Colt Police cartridge, hence the designation New Police, but I may be wrong. I seem to recall reading that Colt refused to mark any of their guns with rival S&W's initials, even for the caliber.
Please note that the .32 Winchester Special cartridge is a rifle cartridge that fires a 170 gr bullet at 2250 fps. I doubt you could fit one into the chamber of your Nagant.