ITV had wanted to secure the trio, but was scuppered by a clause in their BBC contracts that stopped them moving to a terrestrial broadcaster.
Amazon Prime subscribers in Britain, the United States and Germany will be able to watch the show as part of their 79 annual fee and plans are afoot to sell it to other countries.
Explaining the attractions of Amazon, Wilman told Broadcast: "Everyone we have talked to has said to us, 'They leave you alone to make your show'. That's a big one for us - we don't like interference. We don't need to be policed."
The lack of interference particularly appealed to Clarkson, who was regularly reprimanded for his near-the-knuckle jokes on Top Gear.
The format will be familiar to Top Gear viewers, with a mix of studio-based chat and international adventures. Wilman said the three presenters would have "more time to yak" and their friendship would remain the show's central element. "You can't reinvent the sideways, Last of the Summer Wine-type relationship they have. But there will be a new look," he said.
Some Amazon series are made available all in one so subscribers can "binge-watch" an entire series. But Wilman said the new show was likely to stick to the once-a-week schedule.
Clarkson, Hammond and May poked fun at their move away from terrestrial television. "I feel like I've climbed out of a bi-plane and into a spaceship," Clarkson said, while May added: "We have become part of the new age of smart TV. Ironic, isn't it?"