I would wager that "MP" is an ownership mark of the "Metropolitan Police", as suggested by CanTom .... The Metropolitan Police Service was established in 1829 for the Greater London area, and is considered the oldest established police service in the world. The Webley "M. P." (or "Metropolitan Police") revolver mentioned in the thread Tom linked to is actually a specific variation of the RIC model, marked and marketed as such following its adoption by the the Metropolitan Police Service in 1883.
As for the suggestion "Military Police", the absence of any military acceptance or ownership marks suggests otherwise, and unfortunately British Army organization and terminology of the era don't bear it out, either. A very small (21 NCOs and men) "Mounted Military Police" was formed n 1855 for service at Aldershot, which became the "Corps of Military Mounted Police" in 1877. A nucleus of Foot Police was organized for service with the British Army in Egypt in 1882, and from 1885 it existed as a separate "Corps of Military Foot Police" alongside the older MMP. At any rate, based on earlier research, it is my understanding that weapons issued to these units would have been marked "MMP" and MFP", respectively.
As I'm sure you are aware, the mark almost certainly does not signify service with the North West Mounted Police. Their standard-issue revolvers during the relevant time-frame were .450 Mark I and Mark III Adams , and then, starting in 1882, the .476 Mark II Enfield. I have never seen reference to an RIC revolver in NWMP service.