Hiroshima was a fart compared to the average yield of a warhead. Hiroshima was hit by a 13kT TNT equivalent. The average modern warhead is about 300-500kT -- dialble, but assume the max. if it's all in. (That doesn't change that Hiroshima was a tradegy.) A rocket with multiple warheads often enough carries 5*1MT -- all major cities would be rubble.

Ca. 100 modern warheads mean a multi year nuclear winter -- more will be fired, right know more than 1000 active on either side. Even with a small scale catastrophe: if you don't have enough food in your basement, prep. for cannibalism. The civilization would be horrible.