Frankly, I’m astounded. According to the Wikipedia page for Koji Kondo — the guy responsible for the music in most of Nintendo’s major titles, including the Super Mario Bros. theme — hasn’t necessarily admitted that “April” was a major inspiration for him when composing music for Zelda, but it’s hard to overlook the similarities.
Credit goes to NeoGAF’s “Cloudbush” thread, which I’ve written about before. Full of revelations, it is.
And for the sake of comparison, here is a clip of the actual Zelda theme. I felt a guitar version would best demonstrate the similarities. This one is played by Zack Kim, who does it by playing two guitars at once, which is pretty remarkable in an of itself.
Also, I’m going to include a different rendition as well — one of the original Zelda’s dungeon theme. Around the two-minute mark in “April,” there’s a brief part that sounds almost exactly like it, perhaps even more so than the overall song sounds like the Zelda main theme. Here is that, again played on guitar.
EDIT: Okay, while it's not ad admission of anything, Kondo mentioned in an interview with Nintendo Power magazine that he at one point played in a band that covered Deep Purple, among other things. So it at least seems possible that he could have been familiar enough with “April” to have had it influence his work. That's probably as close to a yes or no answer as we'll ever get.
Video games and music, previously: