I hope this is allowed, but what is the transatlantic accent? I've never heard of that

People did talk like that, or at least, similar to that. The TV/movie version of the Transatlantic accent was a bit warped since it was spoken on the screen by actors who didn't actually grow up with the accent in real life, though a few of them did, like Katherine Hepburn.

Old upper class New Yorkers spoke similarly to the "Transatlantic" accent you hear in the movies. Listen to recordings of Teddy Roosevelt or Eleanor Roosevelt, for instance. The accent is dying out if it's not dead already. A couple of the last celebrities to talk with it have died in the last couple decades: George Plimpton and William F. Buckley Sr.

There's also an upper class accent in Boston associated with the "Boston Brahmin" that is also dying out.

A linguist might take issue with lumping those accents together, and see them separately, but they're all related and have seen a marked decline since the early 1900s so that by now they're almost nonexistent.