It's not quite that, anybody in the UK can bring a private prosecution in the UK if the want, so the Post Office was not a special case here. It's really uncommon today, due to the cost, etc. and what often happens is that the "Crown Prosecution Service' which is the government organisation repsonsible for prosecuting crimes normally can 'take on' a prosecution and then choose discontinue it, which effectively stops it. Note that this is only the case in England/Wales, in Scotland the system is quite different. With that said, there are several organisations that do it still - the Royal Society for the Prevention of Cruelty to Animals (RSPCA) still regularly brings private prosectuions.
Historically, all prosecutions were private, and later, the police who investigated them carried them out.