The Wikipedia article is pretty accurate about what they were, but lacks details about how soldiers were selected and how long they might have performed the duties of a regimental police or provost staff.
In my experience each company or squadron within a regiment would provide a number of men for a period of weeks or months to the Provost staff. They would usually be smart, potential NCOs and preferably with good conduct records. Their job was to provide the daytime security staff, mainly manning the main gate and patrolling the perimeter of the barracks, with a separate guard of ordinary soldiers covering these duties overnight, typically 6pm to 6am. The Provost staff would be commanded by a Provost Sergeant. Quite often this might be a senior sergeant with a reputation for steadiness but unlikely to be promoted beyond that rank. He in turn would be assisted by a couple of corporals or lance corporals chosen on the same criteria as the other RPs. In most regiments being selected for the RP staff would be seen as a perk or privilege.
During National Service, when the OP's grandfather was serving, those selected for the RP staff tended to be Regular career soldiers rather than those doing their 2 years national service, since a stint on the RP staff could be a step towards promotion. When a unit was overseas the RP staff might well liaise with the local civil police on matters involving any locally employed civilians working in the camp, especially where petty crime such as pilfering, was suspected.
