The authority responsible for prosecuting the case in court is called the 'prosecutor'. In most cases that will be the Crown Prosecution Service (CPS). There are three types of criminal court in the UK: Magistrates' courts. Crown Court.
Here are steps to follow if you wish to pursue this role: Obtain A-level qualifications or complete the Law Foundation. Complete an LLB degree course. Complete a Legal Practice Course (LPC) ... Undertake specialist prosecutor training.
This includes a three-year law degree, the SQE assessments and two-years of qualifying legal work experience. If you didn't study law you'll need to factor in studying for a law conversion course and/or an SQE preparation course, both of which can extend the training period.
Most criminal prosecutions are brought by the state (eg by the CPS, the SFO etc). A private prosecution is a criminal case which is brought by an individual, a business or an organisation which considers itself to have been the victim of a crime.