Cells have the ability to sense DNA damage. They produce proteins that specifically recognize and bind to various signs of DNA damage, such as single-stranded DNA (normal DNA is double-stranded), DNA ends (which are normally protected inside telomeres), fused base pairs, etc. One possible response is to repair the damage, and this is done quite often. In fact, there is are several checkpoints during every cell division cycle where the cell will pause the cycle and wait for everything to report in as damage-free before continuing. It's actually a bit distressing to realize that pretty much every one of your cells is constantly fixing its genome in real time, and a bit mind-blowing how successful the process is (i.e. the fact that you don't get cancer several times a day).

However, another option in the case that the cell detects irreparable damage is for the cell to self-destruct. If the cell detects excessive DNA damage, it will initiate apoptosis. Also, if the DNA damage is taking too long to repair, the same will happen (kind of like Android's "force close" feature for hung apps). My assumption would be that most of the germline cells that are damaged by the radiation from a nuclear bomb would either fix it or kill themselves, leaving mostly perfectly healthy cells to produce healthy sperm & eggs.

In short, if 10% of your germline cells are damaged by radiation, you will not be producing anywhere near 10% mutated sperm (or eggs).