No all Police (only Service Police not RP) do the Civvy Police course (ACPO/Home Office) because they carry batons and cuffs on UK streets, and potentially could baton or arrest a civvy, it's all to do with liability. The Public Order qual is a different thing and was aimed at places like NI, the PTI school does thier own thing but the only qualified people in the ACPO course are Military Police of any Service.
Batons are offensive weapons so being caught with one if your in a public place (inc in your car) means you will be arrested, unless you carry a warrant card (civvy or Mil)