In the U.S., during the first wave of Beatlemania, so many of their tracks flooded into the American market at one time, it was easy for one particular record to fool listeners into believing they were hearing another Beatles track. I'm thinking of the record, "Lies", by the Knickerbockers. The lead vocal sounded very much like John, and the background vocal shouts ("Yeah, baby!") sounded like Paul. If the Knickerbockers intentionally wanted to sound like the Beatles, they succeeded.
The next best sound-alikes, in my opinion, were the Raspberries on the record, "Go All The Way". Lead singer, Eric Carmen, sounded like a young McCartney, and the song had a Beatlesque "call and response" in the bridge. The bass sounded a bit like early Macca's style.
The Bee Gees vocals on "New York Mining Disaster 1941" was somewhat similar to the Beatles, at least in the two-part harmony. In fact, there were some people who believed it WAS the Beatles, and that Bee Gees stood for BG (Beatle Group). Adding to the intrigue was that the Bee Gees released this single shortly before the Beatles released "Sgt. Pepper's". It had been almost a full year since the Beatles last album (Revolver) had been released, so anything that sounded like the Beatles drew a lot of attention.