13600, 1st of March
On the 27th of October 1962, in our timeline, Vassili Arkhipov refused to launch a torpedo on an American ship without knowing the war hadn't been declared. But in this one, he agreed to launch it, thinking the war had already been declared. Then, the Cuban missiles were launched and so were all the missiles available to any country on Earth, after each country decided to attack its neighbors or its enemies. This catastrophy wiped out 99% of humankind.
In 2405, the shelters that survived are in Tbilisi, Delhi, NYC, Rio de Janeiro, Washington DC, Beijing, Bamako, Bogota, Lisboa, Canberra, Paris, Moscow, Cairo, Doha, Seoul, Ankara, Darwin, Havana, Hô-Chi-Minh, Singapore, Berlin, Nankin, Manilla, Madrid, Ulaanbaatar, Athenes, Dublin, Algers, Cape City, Djibouti, Duchanbe, Ürümqi and Helsinki. They are abandonned by the remaining humans, who start living on the surface again.
The date refered to as "today" will be the first of march, 13600. Note that the Sahara is covered in dense jungles, that Europe is extremely cold and that the planet's magnetic poles shifted.