Where do these words come from? And why are they considered offensive?
Merriam-Webster's Collegiate Dictionary (Tenth Edition, 1993) defines "Paddy" as "Irishman" and adds "often taken to be offensive."
A slew of others, the article notes – The Ninth Edition (1986), the unabridged Merriam-Webster's Third New International Dictionary (1961), the Oxford English Dictionary (1905), the Dictionary of American English (1936) and the Dictionary of American Slang (1967) – mention nothing about “Paddy” being offensive, with the OED tracing “Paddy” back to the year 1790 and defining it as "Nickname for an Irishman” and “pet-form of Padraic or Patrick."
The word “Paddy” started being defined in dictionaries as offensive in the 1980s and 90s, the author explains. As for Paddy Wagon, “a look into the Dictionary of American Slang confirmed that [his] earlier assumption was right: ‘A police wagon used for taking arrested persons to jail. . . . Prob. from the association of there being many Irish policemen.’”
"Taffy was a Welshman" is an English language nursery rhyme with anti-Welsh lyrics, which was popular in England between the eighteenth and twentieth centuries. It has a Roud Folk Song Index number of 19237.