Hardly the equivalent. In 1915, there was no TV, no music on radio, no talking movies, and no equivalent pop cultural explosion.
Virtually EVERYBODY in the Western World -- including my 80-year-old Russian born Jewish immigrant grandmother who never learned to read English -- knew who the Beatles were by the end of 1964, which made them TRUE superstars -- as opposed to today, when the world is full of "superstars" that an awful lot of people -- especially people in their 60s and 70s -- never heard of.
I can tell you though, having been a new teenager when the Beatles first hit the American airwaves, I sure knew who Rudy Vallee was, and recognized the parody when the New Vaudeville Band released "Winchester Cathedral" with the singer sounding like he was singing through a megaphone.
Exactly! I knew that it was a parody of Rudy Vallee too. I was 14 when The Beatles first hit America in 1964, and no one, with the exception of Elvis, was as well known back then. As you said everyone in the Western World knew who The Beatles were.