My guess is that it has been converted to use .32 S&W or .32 Long or .32 H&R.
Not even close. If the Nagant round falls into the chamber, the .32 revolver family will also. The likeliest conversion is probably 7.62x25 as other posters have suggested, but the OP needs to find a round and check for sure, then let us know.
Unfortunately "Buymilsurp" seems to be sold out of cylinders, both original and .32 ACP.
To answer one of your questions, yes the cylinder can be sleeved. The chambers have already been partially bored out, so all it would take to complete the job would be to finish boring them out and then sweat soldering some inserts, ideally in .32 S&W, which would also allow you to use .32 ACP.
I was given a similar Nagant about 30 years ago, ironically also 1925, that someone had crudely drilled out the chambers in an attempt ( my best guess) to allow the use of 7.62x25. I bored the chambers out the rest of the way with a 3/8 drill, and leaving the walls rough, used aircraft structural epoxy (about 30 days to fully cure) to cement the sleeves in place. I shoot .32 S&W Long in it, and it has functioned fine these last 30 years, but the accuracy leaves a lot to be desired. I've yet to try .32 ACP in it, since I have a pretty good supply of .32 S&W, but I don't really see where it would be a problem. The inserts are made from high tensile strength stainless.