Nimoy's Spock is supposed to be the same guy, from the same timeline, that we saw from "The Cage" through to STVI and "Unification". Otherwise his cameo and the whole time travel plot is utterly pointless. Therefore his ship and Nero's Narada are from that universe too - and they had warp drive in 2387, long after Voyager returned from the Delta Quadrant.
Just because the Federation had new technology doesn't mean they share it with everyone else. Also just because there's new technology doesn't mean everyone's going to chuck away their old, perfectly functional warp drives, especially hick miners from the boondocks. Secondly, nobody really knows what Spock's ship was capable of. There's no reason why it couldn't have had a whole bag of tricks including multiple drives.