A public place refers to anywhere that is not a care facility or someone's home. This might be a hotel, a school, sports club or a street. Usually the emergency services, i.e. the police and an ambulance, are called because the death is unexpected.
If the death appears to have been natural but the ambulance service feels that further resuscitation and transfer to hospital is not required, the police will usually arrange for the funeral director working for the coroner to remove the body to the nearest public mortuary (this may be located at a hospital). The ambulance service may do this if the deceased person is located in a very public place, such as a shop.
As most deaths in public places are unexpected, they are reported by the police to the coroner who will usually order a post mortem examination unless a doctor has been treating the person for a condition which might have caused a sudden collapse.
If the death was unnatural, the police will be in charge of the area and will arrange for a funeral director working for the coroner to remove the body to the nearest public mortuary. This may be at a hospital.