I think that the answer lies with P G Wodehouse, who created the characters of Jeeves and Wooster. He began writing this series of books in 1915, using the colloquial language of the era. It's not impossible that 'Pip pip', 'Cheerio, old bean' and 'What What?'' were outdated even back then. But for the generations of Americans who read the books or saw these characters portrayed on the screen throughout the 20th century, they represented archetypal upper-class Englishmen, and it became the stuff of legend that this was how all British people spoke. It's the equivalent of us imagining that all Americans wear cowboy hats and go around saying 'Howdy, pardner'.
I can't say that Americans do that. Maybe a few do because they've read too much P G Wodehouse, as Su.Ki said. The handful who say it are probably just saying it in jest.
In fact, Wooster and Jeeves is the only place that I've ever encountered it.
I'm curious where you heard it and why you believe that "Americans always" say pip pip?
Don't worry about the downvotes. There's no accounting for some people on here. In fact, my comment is sure to attract downvotes.
Personally, I think that IDs should be attached to downvotes with a requirement to explain why that person disagrees. That would go a long way to improving the site, in my opinion.
For what it's worth, I have yet to hear an American pull off any British accent. Usually we sound stupid attempting it. But, as they say: "Imitation is the sincerest form of flattery."
I agree with Su.Ki. about the influence of P. G. Wodehouse. More generally, though, it belongs to a family of comic stage stereotypes. You see them much less often nowadays than before. In the 1950s they were alive and well on TV and in the movies. They were probably in turn based on the stage, what in the United States was called "vaudeville" (and in England "variety.") Anyway, by the 1950s vaudeville itself was more or less gone from the stage but lived on in the movies and TV.
So. Your comic stage stereotyped Frenchman (think Maurice Chevalier) would say "Ooh la la!" and your Scotsman would wear a kilt and carry bagpipes and say "Hoot mon!" and your Englishman would wear tweeds and say "Pip pip" and "Toodle-oo" and "old chap."
Sudeep, please don't try to use any of those words in the list. I posted that link to show what nonsense you find on the internet. It's a list of words which nobody ever uses. I think it must have been written as a joke.