Outside CAS and ATZs wth ATC, the normal Rules of the Air apply and police aircraft have no special priority. ATC may ask someone else to keep clear, but the airspace is uncontrolled. If the situation was as suggested, then ATC did not have control of any aircraft OCAS.
In CAS, ATC operate levels of priority, known in the trade as Categories. AlanM ran through the gist. Because the movements of aircraft in CAS are controlled, it is possible for ATC to exercise their control to give priority to certain traffic.
Following a big incident, a TRA or similar may be invoked. Usually, the police ask for it, but CAA DAP grant the restriction and promulgate it via a NOTAM.
Police pilots are not police officers, but this is not because of any prohibition, but because police officers don't have the minimum flying experience required by the CAA. Some are employed by contractors and some directly employed by police forces as civilian staff. The pilots of police helicopters do not have powers of arrest, but the aircraft is being operated in the service of the police (obviously) and is (usually) tactically commanded by a police officer (observer) who has the normal powers of that office.