It's possible that the Kelvin incident caused the Federation to make contact with groups like the Cardassians earlier than they otherwise would have.
Except, again, it's already canonical fact that there was a Cardassian exile on Vulcan in the 22nd century. So at the time of the first Kelvin movie, the Federation would've been aware of Cardassians as a species and culture for nearly a century, even if it didn't have regular or formal interactions with the Cardassian state.
Also, cultural exchange can be indirect. Maybe the Federation got Cardassian beverages from a third civilization that traded with both.
They apparently made contact with the Romulans, which would explain how Uhura could know their language, and how Starfleet was aware of the Vulcan-Romulan connection.
"Balance of Terror" established that the peace treaty a century before was negotiated over subspace radio. Therefore, at least one side must have known the other's language -- probably both, so they could verify the accuracy of the treaty's translation. Even without face-to-face contact, they probably would've intercepted and decrypted each other's transmissions, salvaged data banks from destroyed ships, etc. and learned each other's languages that way.
As for the Vulcan-Romulan connection, the
Kelvin crew themselves presumably discovered that when the
Narada attacked them. Or maybe the attack prompted renewed scrutiny toward the Romulan Empire (since the UFP wouldn't have known the ship was from the future) and the Romulans came out of isolation to deny being culpable.