Beatrice Stockly, who was previously abducted by Islamists in northern Mali in 2012, was taken for the second time on January 7 by armed men who stormed her home in the country's fabled city.
The eight-minute video includes triumphant montages of jihadists brandishing weapons and a masked English-speaker who claims responsibility for the abduction on behalf of Al-Qaeda in the Islamic Maghreb (AQIM).
"Beatrice Stockly is a Swiss nun who declared war against Islam in her attempt to Christianise Muslims," the speaker said.
Stockly has previously been identified as a missionary and social worker in her 40s who had lived in Timbuktu for years.
With at least three other fighters dimly lit in the background, the speaker said AQIM carried out the abduction and listed conditions for Stockly's release.
They include the release of a number of AQIM fighters jailed in Mali and one of their leaders, Abu Tourab, detained at the International Criminal Court at The Hague.
Abu Ahmad Tourab is the nom de guerre of Ahmad al-Faki al-Mahdi, who is accused of ordering the destruction of historical monuments in Timbuktu in 2012.
Tourab, who was a leader of the Al-Qaeda-linked Malian group Ansar Dine, is the first jihadist to appear before the war crimes court. Stockly appears at the end of the video dressed in a black hijab.
She identifies herself and the date as Tuesday, January 19, 2016. A Swiss foreign ministry spokesman told AFP that Bern was "aware of the video in question," and called for the hostage's unconditional release.
The foreign ministry has previously said that it discouraged Stockly from further travel to Timbuktu following her first kidnapping.
The security situation in northern Mali has not stabilised since a loose coalition of Islamists and Tuareg rebels overran the region in 2012.
Jihadist fighters were largely chased from the area in 2013 by a French-led military intervention, but entire swathes of the area remain beyond the reach of both the Malian army and foreign troops.
The speaker in the video said AQIM had developed an expertise in dealing with Western hostages and was prepared to be "patient" while waiting for its demands to be met.