Whilst some personal service records of regular members of the Metropolitan Police are held at the Public Record Office, those of the provincial police forces are not held there. I'm unable to say whether the personal records of the Metropolitan Special Constabulary still exist, and if they do, where they are held.
Insofar as the records of the provincial forces are concerned. These are usually (but not always) held at the County Record Office. That said, records of the Special Constabulary and its members were often not retained and were destroyed following a suitable period after the individual ended his service.
Your medals were awarded (post August 1919) during the reign of George V. At that time in England, Scotland and Wales there were approximately 200 separate police forces. All those forces maintained a Special Constabulary, whose members would have been entitled (subject to service conditions) the award of the medal. So, there is no easy way of tracing the force the recipient belonged to.
Best of luck.
Dave.