We could kill all of the large (insects and up) life no problem. There are about 30,000 nuclear weapons in the world today, so in what follows I’ll assume the worst case scenario; that all of them are evenly spaced across the Earth’s land masses and set off. So as long as there’s liquid water somewhere on Earth (even ultra-high pressure acid water) there will almost certainly be life. And of course, the world has never been destroyed by an all encompassing nuclear disaster. However, we have fossil evidence of microbial life dating back about 3.8 billion years, and the moon’s marias were still being created (by really, really big impacts) until about 3 billion years ago.