US policing evolved from slave patrols used to catch and kill runaway slaves.
Slaves called the slave patrollers, "Patty Rollers" or "Rollers." Black folk today still call cop cars "rollers." Slaves called the wagons they used "patty wagons."
Many slave overseers were Irish. When early police forces formed, many became cops. And Irish were over-arrested for alcohol.
British folk called Irish folk the anti-Irish slur version of paddy, hence "paddy wagon."