The deal is a coup for Amazon, although the presenters were hamstrung by the fact a clause in their old BBC contracts prevented them hosting a car show for a UK television channel for the next two years. That clause scuppered their plans to move to ITV. Netflix were also said to be interested in signing them, but Amazon secured their services.
Subscribers in the UK, US and Germany will be able to watch the series through Prime and plans are afoot to sell the show to other countries.
How the trio reacted on Twitter (app users tap here)
Explaining the attractions of Amazon after years with the BBC, 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."
In a separate interview with Radio Times, he said: "No-one telling us what we can and can't do, just us hopefully producing great programmes - it feels really liberating."
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 because that is outside of television, that is theirs. But there will be a new look, new elements, new home," he said.
Some Amazon series, including the comedy drama Transparent, are made available all in one go so subscribers can "binge-watch" an entire series if they choose. However, Wilman said the new show will likely stick to the once-a-week schedule familiar to Top Gear's BBC Two viewers.
"We're a one-trick pony. I wouldn't watch 12 episodes in a row," he joked.
Asked if he would like the show to go head-to-head with Top Gear, Wilman admitted:
"The child in me probably would," but said there was room for two motoring programmes.
Jay Marine, vice president of Amazon Prime Video EU, said: "Customers told us they wanted to see the team back on screen, and we are excited to make that happen.
"Millions of Prime members are already enjoying our ground-breaking original shows. We can't wait to see what Jeremy, Richard, James and the team will create in what is sure to be one of the most globally anticipated shows of 2016.
"This is a golden age of television, a great time for TV makers and storytellers. Our approach is to give programme makers creative freedom to be innovative and make the shows they want to make. This is just the start, you should expect to see more world-leading talent and the biggest shows on Prime Video."
