Y'all have to remember that the Klingons had possession of the Narada and had held Nero and his crew prisoners for 25 years.
That will give the Klingons plenty of time to figure the Narada access codes and download all the information that they had found.
Also plenty of time for federation spies to steal that same information from the klingons.
The Dreadnought most likely was built from information that was gotten from the Narada. Which mean, that the Dreadnought is not from the future, but is base on future technologies.
See, again, I don't know if we can really canonize a deleted scene... but even if we do, we really don't know what the Klingons had, they may have attempted to capture the ship, but only succeeded in capturing Nero and some of his crew, the rest of the miners taking the Narada to who-knows-where. Nero may have even given himself up to the Klingons specifically to allow the the Narada to escape and wait out his time, saying nothing, knowing that when they do escape, the Narada is waiting for them. All the while, the Narada's supplemental crew trying to survive for 25 years waiting for Nero to return... it opens many possibilities for future tech seeding the known galaxy.
I even wonder if that was the writers plan all along, to not only re-set and alter the TOS timeline to fit a cinematic universe, but introduce 24th century tech in the mix to upgrade that universe's technology and ship design to meet modern audience standards.