The Bhakra and Nangal are two separate Dams. Bhakra Dam is the main Dam that has a reservoir named Gobind Sagar. The capacity of this reservoir is 9.34 billion cubic meter of water, It is one of the most prestigious projects from the Indian Government whose inception was done even before Independence around 1945-1946. These two Dams were build over Sutlej river to control floods in these mountaineous areas.
Now if you note the capacity of this reservior it is the thrid largest reservior in India and has the potential to practically wash out the city of Chandigarh, Ludhiana and Parts of Haryana and Delhi. It has 4 canals in radial orientation originating from Nagal dam which is the next dam after Bhakra dam (moving from east to west).
The purpose of Nangal Dam is to stop the sudden rush of water from Bhakra Dam if it happens, so that there is a chance to buy more time to evacuate important cities. Thus if only Bhakra dam fails, the Nangal dam can control the severity of the damage to a major extent.
Now if you ask what if both dams fail back to back: If this happens then several low lying areas especially from Punjab and Haryana will get flooded. And from Punjab there are 5 rivers and the land between these rivers is quite plain and thus these two states will be hit severely. Major cities would include Chandigarh, Ludiana, Ferozepur, Rupnagar, Anandpur Sahib, Kalka and Ambala City.
Apart from physical damages the states of Himachal Pradesh, Punjab, Haryana, Rajasthan, and Gujarat have stakes in Hydro electricity generated from these dams which will also affect. Northern Grid will also be affected. Upto what extent I have no idea to comment. Other minor industries such as tourism will also get hampered.
This is what I could gather so far. Wll keep posted as I get more. My sources have been wikipedia and google maps only and this is my estimation based on certain logical approach.. 🙂