There have been multiple Earths in which Superman was raised by the Waynes.

The first was a dimension-travel story where Batman visited a world in which Superman is secretly Bruce Wayne, and the crime-fighting partner of Robin; crime photographer Vicki Vale is a perfect look-alike for Lois Lane; there is no Batwoman; James Gordon is editor of The Daily Planet; the Man of Steel suffers from a weakness to Thilium rather than to Kryptonite; and Freddy Forbes was a television comedian who looked exactly like the Joker. In the story, Batman helps Superman capture Red Raven and his gang. Designated as Earth-136 by John Wells.
World's Finest #136, "The Batman Nobody Remembered"

The second was an Imaginary Story in which Kal-El's rocketship crashes near Gotham, is found by a young James Gordon, who hands the baby over to the Waynes to raise (who have no children of their own). The Waynes are not killed, so Bruce grows up to be Superman. Bruce marries Barbara Gordon who has convinced him to give up crimefighting to work as a scientist. While he's working on curing the common cold, the Gotham underworld starts a crime wave and in the midst of it, Commissioner Gordon is killed. Superman tries to go back in time to save his father-in-law but, alas, according to pre-Crisis time travel rules, one can't appear twice in the same time, so all he can do is helplessly watch the murder as a ghost. Barbara, realizing that with Superman gone, Gotham does need a protector, and Barbara decides that she will be that protector - "This is a job," she tearfully tells Bruce, "for Batwoman!" She manages to capture the criminal who killed Gordon and decides to continue as Batwoman. Designated as Earth-353 by John Wells.
Superman 353, "The Secret Origin of Bruce (Superman) Wayne"
Superman 358, "A Day in the Life of Bruce (Superman) Wayne"
Superman 363, "A Night in the Life of Bruce (Superman) Wayne"

Note: an amalgamated version of these first two worlds might lend us a Batwoman & Robin team, possibly a less dark, pre-Crisis version of the world depicted in Thrillkiller, where there is also a female Joker named Bianca Steeplechase. Post-Crisis, she kills Robin in 1961 and policeman Bruce Wayne is inspired to become Batman and join Batgirl in 1962, although that would plainly be noncanonical here. Designated as Earth-61 by John Wells.

The third was another Imaginary story in World's Finest #167, "The New Superman-Batman Team." Kal-El grew up powerless due to his rocket passed near a Gold K meteor. Lex Luthor invented a serum which gave him super-powers, allowing him to become Superboy, and later, Superman. When the Kents were killed by a robber, young Clark was left to be raised by his wealthy uncle Kendall and devote his life and fortune to fighting crime as Batman, unaware of his Kryptonian , Clark eventually learned of his heritage after Supergirl's arrival on Earth. The adult Lex had brown hair and wore glasses, and became a Daily Planet reporter. When sent to interview his boyhood chum, millionaire Clark Kent, Lois Lane tagged along and fell for Clark. Eventually, to save Clark's life, Lex had to transfer his own powers into Clark's body, and the remnants of Gold K radiation into his own... after which he voluntarily left the planet in a spaceship so that his presence wouldn't endanger Clark or Kara. Clark ended up as Superman, and Batman simply disappeared. Designated as Earth-167 by John Wells.

Finally, in 1993's Speeding Bullets Elseworlds title Kal-El's ship is found by Thomas Wayne. After his parents are murdered, he becomes a Batman who can fly, and more. He later encountered Lex Luthor as The Joker. Lois Lane helps Kal realise his turn to violence is wrong, and he winds up as Superman yet again. Designated as Earth-1993 by Nathanial Parkson.

Also worth noting is Action Comics #314, "The Day Superman Became the Flash." In a computer projection, Jor-El sees Superman become a costumed detective on a world of night.

Thanks to Bob Buethe for the information on WF #167 and Terence Chua for the information on Superman #353.