In the 1800s the U.S. upper classes looked toward England and aspired to British high culture. Wealthy people sent their children to private "prep schools" that prepared them for Harvard or Princeton. These prep schools and colleges actually taught something resembling British English, the once-famous "Harvard accent." The "Harvard accent" still existed in the 1960s.
This may be the reason why, when people in the U.S. think of "the English," we tend to think of the British aristocracy. Our images seem to be come mostly from the days 1900-1914, just before World War I.
The writer P. G. Wodehouse wrote a series of very funny comic novels starting in 1915, about a foolish British aristocrat named Bertie Wooster and his wise butler Jeeves. These novels perpetuated a British stereotype. Say "Englishman" and many in the U.S. immediately picture someone like Bertie Wooster.
We also picture someone like Professor Higgins in the musical "My Fair Lady," based on Shaw's 1912 play "Pygmalion." Agatha Christie's mystery novels, set in the 1920s, also involve rich aristocrats living in big houses.
So, the STEREOTYPE is that the "English" are ALL rich aristocrats, who are "stuck up," haughty, and arrogant. They all live in gigantic stone mansions like Downton Abbey. They all go around saying "Pip pip!" and "Cheerio!" and "Toodle-oo." They always dress formally. They all have butlers. Because of Sherlock Holmes, we also think everyone in London drives around horsedrawn hansom cabs over cobblestones. London is always shrouded in thick pea-soup fog. And all of the differences in British speech are viewed as affected.
I'll leave it to a Brit to describe the British stereotype of the United States.