We don't have a lot of overt evidence (e.g. man/woman in the street talking about it) to go on, but by the time Star Trek (2009) is over:
- The Klingons have known for 23 years, because they've had custody of Narada all that time and have been mining it for secrets.
- The fact that the Enterprise is more advanced in this timeline when it's finally built (late) suggests that the Federation has been either frantically researching or stealing from the Klingons comparable advances in technology mined from Narada, which suggests that someone in the Federation (Section 31?) already knew about the ship from the future, if not all the implications.
- Spock's re-emergence, the destruction of Vulcan, and whatever report Pike would have filed after his discussions with Nero would have given away the rest of that part of the story. Ambassador Spock's active involvement with New Vulcan means that at least Vulcan leadership knows, and with only 10,000 Vulcans left at all, it'd be fair to assume that other Vulcans know as well.
- Romulan leadership may or may not know; Romulan control over its populace probably means its people don't unless they feel they ought to. In the Prime Timeline, the Romulans were still quiescent at this point (2260) and in fact, no human knew what a Romulan looked like until 2266. In the Kelvin Timeline, though, nobody seemed surprised that Nero and his crew were Vulcanid, so this is clearly not true. That said, Star Trek: Enterprise implies that Romulans were actively spying on the proto-Federation, and it's safe to assume they spied (in both timelines) on the Federation pre-2266 as well. They also would have been foolish not to be spying on the Klingons. It's all speculative, but I think we can assume that Romulan leadership knows about Narada, and the future destruction of their planet.
- We can also speculate that the Federation might actually have reached out to Romulan leadership to warn them, "Hey, 120 years from now or so, you're screwed. Since we know now, in this timeline, that it's going to happen, you might want to think about relocating, now. Just saying." There's a lot of reason for mutual suspicion there, but Federation ethics would probably leave them feeling compelled to try.
What's totally unclear is how much the general Federation public knows. The Federation is often glossed as having a fairly open press culture akin to the US and Western Europe, but this would be a pretty big bombshell to drop on the general public. On the other hand, if Ambassador Spock is publicly active with the New Vulcan project, it might be hard to hide it, given that he doesn't appear to be even trying to operate under a pseudonym (a la the TAS episode "Yesteryear").