Leading Seaman Turney, 26, said they had been seized in the Gulf because "obviously we trespassed" in Iranian waters - something the UK disputes.
She said her captors were "friendly", "thoughtful" people and she and her colleagues were unharmed.
The Foreign Office said the footage was "completely unacceptable".
Earlier Iran said it would release Leading Seaman Turney "very soon".
Iranian Foreign Minister Manouchehr Mottaki said she would be released on Wednesday or Thursday.
Earlier the UK said it was suspending bilateral contacts with Iran amid the dispute over Friday's incident in the Gulf.
Eight sailors and seven marines were taken at gunpoint by Iranian Revolutionary Guards.
They were seized after they had searched a merchant vessel.
'Issue under consideration'
Iran has insisted the group, based on HMS Cornwall, which has its home port in Plymouth, were in its waters when they were taken.
Speaking on the sidelines of an Arab summit he was attending in Saudi Arabia Mr Mottaki said: "Today or tomorrow, the lady will be released."
He later told reporters: "The issue is under consideration. I think it will be solved soon based on rules and regulations. I think I have informed you that the lady will be released very soon."
His comments came shortly after Tony Blair said it was time to "ratchet up" pressure on Iran.
Earlier, the Ministry of Defence issued data it said proved the group had been 1.7 nautical miles inside Iraqi waters when they were seized.
Deputy Chief of the Defence Staff, Vice Admiral Charles Style, gave detailed co-ordinates which he said proved that.
This is a copy of a letter e-mailed to The Associated Press by the Iranian Embassy in London, Wednesday, March 28, 2007, allegedly written by captured British sailor Faye Turney to her family.
The co-ordinates were 29 degrees 50.36 minutes north, 048 degrees 43.08 minutes east.
The MoD backed up its assertion by releasing a photograph of a handheld global positioning satellite device in HMS Cornwall's Lynx helicopter as it flew over the searched merchant vessel, confirming its position.
Vice Admiral Style said the sailors had been "ambushed" and their detention was "unjustified and wrong".
The UK government said the Iranians initially said the merchant vessel had been at a point within Iraqi waters, before later providing a second, alternative position, within Iranian waters.
Iran's embassy in London issued a statement in response to the UK data, in which it said the sailors and marines had been
The statement, quoted by the official IRNA news agency, said "the governments of Iran and Britain have the ability to solve the incident through contacts and close co-operation".
The BBC has been told the group are being held at an Iranian Revolutionary Guards Corps base in Tehran.
Güncelleme Tarihi: 20 Eylül 2018, 18:16