In the novelisation of the first StarTrek-movie you could read, that "For nearly one century, the first quantum transition of a spaceship into the Warptechnology ensured for the fact, that all earlier theories which were based on too narrowminded interpretations of Einstein's work, were nonsense". (p.94)

Einsteins work? Right, we are talking about the humans from earth. If we read the "nearly one century" as "between 89 and 99 years ago", we can count back from the Vīger-Incident, which was sometime between 2270 and 2272, until this first Warp-Engine.

Therefore the first Earth-Warp-Ship would have been built between (2270/2272-89/99=) 2171 and 2183.

A Ship called "Bonaventure" (Most fan sources have given the registry as NCC-S1100 but clearer screen shots show the registry "10283NCC") appears in the animated episode "The Time Trap" and is described as "the first ship to have warp drive installed". She is stuck in the Delta Triangle and Spock says that "The crew's descendants may still be living," which places her disappearance at least several decades before TAS.

TAS must play before STM (sometime between 2270 and 2272) and after TOS second season (2266) and third season. We don't know when the Bonaventure get's lost; we don't know the exact date from "The Time Trap"; we don't know how long the humans of the future with their medicine could live. But there seems to be no error in the timeline.

It was thought, that "Cochrane has been dead for 150 years". The "Companion" intercepted his disabled ship, brought him to a planet, and rejuvenated him. Cochrane had set out in a ship at the age of 87 to die in space, as he told Kirk in "Metamorphosis", TOS 2. season, which took place 2266.

We can read above "150 years" as "between 130 and 170 years". Cochrane must have been born (130/170 years + 87=) 217/247 years ago; that would be between 2018 and 2049.

Cochrane "died" therefore between 2105 and 2137 in space. As we can see for sure, Cochrane did not build the first Warp-Ship on Earth!

But Cochrane looked at Spock and wanted to say something about the Vulcans - as we can see, he still knew the Vulcans! Maybe he knew them by Subspace-Video-Communication? And Earth learned from his Warp-theory the same way. Or the humans from the Sol-System had contact with an alien race, which traveled between the stars. And they heard about this theory. And sometimes later they build their first ship. But the human design was better - and they organized the "Starfleet" for the new Federation of Planets?

However, Cochrane should be at least 20 years old (after 2038) and should develope his Warp-Theory before his "death" at the age of 87 (see above). Therefore Cochrane's step to Warp-Drive would have happend between 2038 and 2137 for the Alpha Centauris.

The United Federation of Planets (or UFP) was founded in 2161 [TNG: The Outcast] and the "Uniform Code of Justice" was signed. ("The Drumhead" [TNG])

It was signed on Babel; maybe simple Warp-ships were already in use. They were used by humans from Earth - but at first it was not a design from earth. Or they used sleeper-ships or impulse-drives or "atomic propulsion" (as the Elasian ship from TOS: Elaan of Troyius). .