The footage, shot on 19 September, shows a Russian helicopter hovering over the deck of Greenpeace's Arctic Sunrise.
Armed men drop on to the deck, while members of the Greenpeace crew are seen with their hands in the air.
The footage also shows the ship being towed towards Murmansk, where the 30 people on board were taken ashore and arrested.
The families of all six UK nationals being held by the Russian authorities following the arrests will meet a government minister for the first time on Monday. The meeting will be with foreign minister David Lidington, minister of state for Europe.
Prime minister David Cameron has called the charges "excessive" and urged president Vladimir Putin to make sure those being held can come home.
The prime minister said he had appealed to the Russian leader to "de-escalate" the situation.
Cameron told BBC Radio Devon: "They are not hooligans, they are protesters" although he acknowledged countries needed "tough rules" to protect oil platforms.
He said: "Their charges have been dropped from piracy to hooliganism, but I still think that is excessive. They are not hooligans, they are protesters.
"I totally understand that countries have to have some quite tough rules to stop people invading oil platforms, but I have appealed to Vladimir Putin to try to de-escalate this and make sure that these people can go home."