modern: Van or truck used for transporting prisoners.
Origins of the phrase are not sinister. The origin is from the era of horse drawn box enclosed wagons employed by the Police Department, especially of New York City. These wagons had the sole identification painted on the sides, in large letters the initials P.D. When spoken the term "P.D. Wagon" easily becomes "paddy wagon".
While the word "paddy" may be in modern use in derogatory ways, the development of those meanings are separate and unrelated to the "P. D. Wagon".
These derogatory uses however, have led to fabricated, invented and much more complicated explanations of the phrase "paddy wagon" and while a word or phrase may develop new meanings over time, and thus become valid of themselves, that does not change the true and simple origin as a slurring of the phrase "P.D. Wagon"