Old Bill" was King William IV, whose constables were an early form of police. (It is often said erroneously that he was on the throne when the police were founded. Actually he did not succeed George IV until 1830).
A cop is a slang term used to describe police, constables, sheriffs, prison guards and other law enforcement officers. The usage of the word "cop" to refer to law enforcement may have been derived from the verb cop, which means to take or seize and most likely does not come from the British reference to Constables On Patrol.