Ingame from a note in metro exodus from an old woman who surivied the blast but latter died from the radiation she said after the bombs she couldent see much at first and then when she looked out the window after gaining her sight back she said the city was covered in a green fog with the sirens blaring and no life at all to be seen, after that she felt sick and died.

In books a woman survives in a currency exchange booth in the vestibule of a metro station and says that there was still bombs and exlosions going on for a good while in some parts of the city, she also saw a bunch of dead bodies at the entrance and cross the place, latter she said some dogs came and ate the bodies but also a tank drove by but she couldent get outside to it in time. Later she also saw a burned man who went down the steps to the station and was seemingly let in while she was left outside with no answer when she tried to get in.

Ingame and book lore it seems the nukes caused an EMP blast disabling all electronics causing planes to fall out the sky and stuff but in real life testing most vehicles will not be affected adversely by EMP's as the US tested 50 popular cars and something like only 2 stopped working permanently and needed to be fixed up while the rest restarted with out issues but during the EMP the lights,radio and other electronics started to play up with a mind of their own but i imagine a nuclear EMP is probably strong enough to fry anything not protected.

In Moscow also the nukes where mostly shot down by the missile defense system meaning while the city wasn't turned into a glass crater the city was showered with radioactive particles and a few nukes did go off by the looks of things in flashbacks so the air would likely be full of dust and debris.

If you have the stomach for it look at the Beirut blast videos to see the aftermath, the explosion ripped the roofs of buildings, blew in windows and pealed the plaster right off brickwork. now imagine that but worse for a nuclear situation.