There's an Enterprise episode where Archer notes that Cochrane gave a university commencement address where he told "the real story" of first contact, complete with the time-traveling cyborg zombies from the future, which he later disclaimed. T'Pol indicated that the prevailing historical view was that Cochrane had a brief relapse with his alcoholism before the speech, and was completely drunk at the time, leading him to spin a tall tale to amuse the students.
Aside from Cochrane's little indiscretion described above, I'm sure most of the people involved kept their mouths shut about time-travelers from the future, either out of fear of the consequences of insisting such an outlandish story was true (wouldn't the Vulcans have seen another ship in orbit, or detected a space battle? And haven't the Vulcans conclusively proven that time travel is impossible?), or out of civic-minded interest in ensuring that the future that saved their bacon will actually come to pass.
And Data's head is actually a bit of an outlier in being an artifact from a future timeline that actually comes to pass. Most of the time someone travels back to the present in Star Trek, they ensure their own future doesn't come to pass, but whatever they leave behind doesn't just disappear, back-to-the-future style. There's future-Alexander trying to turn his childhood self into a Big Man so he can save Worf's life years later, Future-O'brien who replaced his earlier self, Admiral Janeway who got Voyager to Earth decades ahead of schedule. Characters being aware that they're going to go to the past in the future, and then actually doing so, is actually pretty rare on both sides of that equation.
Interestingly, one of the Shatnerverse novels postulated that the point where the Prime Universe diverged from the Mirror Universe was that, in the Prime Universe, Cochrane didn't warn anyone about the Borg, but in the Mirror Universe, he did, leading to decades of militarization and fascism to prepare for the inevitable Borg invasion. That was always a bit of a stretch to me, though, and was ruled out by later canon.