This is what I imagine it is like to be fanatical Beatles collector in the year 2518 (give or take):
There were digital archives of media from Earth-That-Was that came with the initial colonists. Those archives have since been damaged, lost, found, damaged a bit more, lost again etc… Some things might have been better preserved than others. Some things might have had more "staying power" in the colonists' consciousness and thus were harder to lose as often.
So, Lennon heard "Across the Universe" in a museum not necessarily as a "Beatles song" but as an old Earth-That-Was-song that seems to relate to space travel and thus picked by the curator as historically relevant to an exhibit on the birth of space travel. And then Lennon did the research to find out who that was and was there more of it to listen to. Some of the Beatles catalog was probably quite easy to recover and other songs not so much. He has shelled out money to hackers to find songs rumored to exist, but that aren't easily available - like modern collectors do. (For instance, Lennon has cover versions of the song "Yesterday", the most covered song in history, but has never found the actual Beatles version. It's like his Holy Grail.)
I'm sure Lennon looks for other things too, like memorabilia, the Beatles' movies, photos, solo works, etc. I'm also sure that other popular music survived in a similarly limited fashion.
- So the end result: Are people who've never heard Lennon's peri-operative ramblings going to know who John Lennon is? Probably not. Will they know a Beatles tune if he sang or hummed it? If it's something like "Hey Jude" or "Let it be" then they just might, but it would be like Greensleeves or Christmas music to them.
