Currently there are 12,331. These weapons are divided up among many nations, and only a fraction of them are actually "ready to launch" at any given time. If launched most of them will be targeted at military targets, which are often located in remote places - silos in the middle of nowhere, carrier groups out in the ocean, forward military bases or stockpiles, and so forth. They wouldn't be fired with intent to "wipe out" humanity. There would be entire continents that nobody bothers firing at - why waste precious nukes on countries that are uninvolved in the conflict?

Nuclear winter is no longer thought to be as bad as the most extreme predictions from back in the 1960s. And even with those extreme predictions it still wouldn't lead to human extinction. Humans are an incredibly robust species. We don't need infrastructure to survive in harsh conditions. Inuit survived in the arctic for thousands of years without anything fancy, and you're not going to see conditions that harsh everywhere on Earth regardless.