I have no proof of this, but based on the context, I’d say it’s one of many songs on the “Bridge Over Troubled Water” album that deals with the slow breakup of Simon & Garfunkel.
Despite the great success they’d had, artistically and commercially, Paul Simon and Art Garfunkel had a lot of vehement disagreements. Both were talented guys with healthy egos, and that often led to tension. And it didn’t help that Paul Simon was so slooooooow, when it came to writing songs. To some, that meant he was a perfectionist, to others it meant he was just lazy.
Regardless, during the long periods when Simon wasn’t writing any music, Garfunkel started pursuing other interests, like acting in movies. In spite of the fact that Simon had often treated Garfunkel like an unnecessary nuisance, he was genuinely hurt by Garfunkel’s lengthy absences. It became pretty clear that the two were drifting apart, and headed in different directions.
Several tracks on “Bridge Over Troubled Water” reflect that. “So Long, Frank Lloyd Wright” for instance (it helps if you know that Garfunkel was once an architecture student). The lyrics to that song say, essentially, “Bye Art, we had a lot of great years together, and I’ll always treasure the great times we had and the great music we made together.”
Then there’s “Why Don’t You Write Me?”, a song that practically pleads, “Come on, Art, where are you? Come back!”
“The Only Living Boy in New York” isn’t anywhere near as straightforward in its lyrics as those other those other two songs, but I think the meaning is similar. When Simon writes about “Tom,” I think he’s probably talking about Art Garfunkel… you may remember that the two of them originally called themselves Tom & Jerry. I THINK the song is another plea, along the lines of, “Come on Art, I’m here in New York waiting for you. Come back, I KNOW you’ve got to be ready to sing with me again.”