It shows May driving with a broken arm, Clarkson driving on the Game of Thrones set and Hammond waking up in a buggy to find it has been hoisted into the air.
The footage was shown to delegates at the Edinburgh Television Festival.
The Grand Tour producer Andy Wilman said: "Like them or loathe them, they are still doing their thing."
Wilman, who worked with the trio on Top Gear, followed the presenters from the BBC to Amazon after Clarkson's high-profile altercation with a producer last year.
Screening a montage of clips at the festival, he said: "This is not a trailer but I put something together to show we have been busy. We have been going around the world with big ambition in the films."
In the footage, May could be seen with his arm in a sling, and Wilman revealed: "He fell over the night before filming, coming out of a pub.
"He rang me up and said 'I've broken my arm'. His car was automatic so we stuck him on Eurostar, pumped him full of drugs and left him to it.
"He did more damage to it because he had to keep on going but he's falling to bits anyway. We are not in Usain Bolt territory with that body."
The footage also showed the trio racing Maseratis, a Rolls Royce and a Porsche, as well as Clarkson on a jet ski.
Radio Times reporter Jonathan Holmes wrote afterwards: "From this short tease, The Grand Tour looks like the biggest, brashest, loudest, most extreme version of Top Gear imaginable."
The first episode will be 70 minutes long and the team have signed up to make 12 shows a year for three years.
Wilman said everything was being filmed in 4K and that 90% of the pre-recorded films had been shot. But he rubbished press reports that each episode had a £4.5m budget.
The furore that led to Clarkson's sacking after punching a producer during an argument over hot food "was a perfect storm that was coming", Wilman explained.
"[Top Gear] got bigger and bigger by accident. We never adjusted to that and were collapsing under the weight of the work we were doing.
"We had [things] like Argentina go wrong, so it was all building," he added - referring to an attack on the Top Gear team in the country during filming in 2014.
Locals threw rocks at crew members' cars in an apparent protest at a number plate which appeared to refer to the 1982 Falklands war.
"I'm speaking as someone who loves the BBC and there were a lot of people who were great with us and some people there weren't great with us and didn't want us there, so it became a battle," Wilman said.
"It became personal and confrontational and when everything went [wrong] in March, that was critical because it was going to be a victory for somebody.
"It wasn't going to be a resolution because I think some people didn't have the will to make it work on the management side, and I didn't have the maturity to make it work either.
"Everyone had taken their position, we were all entrenched."
He added: "It was sad... but we were to blame too, I was throwing my toys out of the pram, I was vicious in my reaction to everything and it became thumping heads."
Clarkson was replaced by former Friends star Matt LeBlanc and BBC Radio 2 presenter Chris Evans, who resigned in July after his first series.
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