It's hard to say for certain without knowing more of the state of the world.
Cochrane and Sloane would've both been on the test flight (and the third seat would probably also be a critical scientist/engineer on the project). They wouldn't have been the only experts on the project, but they were almost certainly the most knowledgable about the ship's construction and the theory behind the warp engine. Even with their notes, designs, and schematics, whoever was still in Montana would have no way to investigate the accident beyond whatever telemetry they happened to receive before the malfunction, and they would need to start over from scratch to build a new test vehicle. Even if there was another suitable missile at the complex, they've got to built the warp engines and reactor, as well as the crew capsule. So, if everything goes right (the people on the ground have the expertise to rebuild the ship, they have enough information to avert what destroyed the Phoenix, and they have or can collect the necessary materials), we're still talking years before another attempt.
And then there's the rest of the world. The world's infrastructure has been shattered, but it's likely plenty of places are better off than the encampment we saw outside of Bozeman, and would've had the capability to detect a missile launch and explosion, so it's entirely possible someone shows up, either what remains of the U.S. government/military or from outside the country, to secure the site, and it's even money whether they consider the research valuable and take it to develop themselves, or decide it's crackpot nonsense and just throw the whole thing away.
But that's not to say the warp engine never gets built. There are some ideas and inventions that can just be spontaneously thought up, while others (more common in this age of rapid development) are inevitable confluences growing out of recent inventions and discoveries. Look at the famous story of the telephone being patented almost simultaneously by two independent inventors, or the number of people working on a powered flying machine at the turn of the 20th century. It's hard to say which of these warp drive is (were Cochrane's theories developed and published before the war? Were they based on new discoveries that were?), but either way, I'd say it's a safe bet that the scientific and technological infrastructure existed, and Cochrane didn't leapfrog hundreds of years of experimentation and theory by landing on the right solution of his singular genius.
So, I'm guessing another warp engine is built on Earth within fifty to a hundred years, tops. Now, things could get very ugly depending on who owns it, but let's assume it's the "good guys," and access to the technology is pretty equitable. A lightspeed engine isn't going to open up the universe to you, but being able to get anywhere in the solar system in less than a day is going to change things up. Distant asteroids and moons are as quick to travel to as cities on Earth, giving humanity a ready supply of precious metals, rare isotopes, and other space resources. The solar system probably urbanizes overnight, and humanity becomes very easy to notice. The Vulcans seem to pass by at least once every hundred years or so, but maybe it's another species who happens to be the first to pass by in Earth's burgeoning interplanetary age. The Andorians, or Denobulans.
