Avogadro Corp is my answer to this question. Alas that nothing better exists yet. But it's surprisingly prescient considering that it was written in 2011!
It's not exactly brilliant literature, it reads like a big-budget Hollywood action thriller. But it's got a lot going for it in the realism category relative to anything else I've seen.
One answer is that it might be soon enough that it's very much like right now, and so any contemporary fiction could do. Another is that "the future is already here, just unevenly distributed", and extrapolation can give you credible stories (so long as they're not set too close to an AI lab).
On longer-timelines views, a prerequisite to writing such a novel would be accurately forecasting the trajectory of AI research and deployment right up to the end. This is generally seen as rather difficult.
The novel After Life by Simon Funk has quite a few flashbacks to the world prior humanity's end, though it takes more than a year. I find it one of the more hopeful stories in the genre.
The Sprawl trilogy by William Gibson (starting with Neuromancer) is basically about this, and is a classic for a reason. It's not exactly hard sci-fi though.
Does anybody know of any kind of novel which is set in a plausible version of the world a year or so before humanity id ended by AI, at least as we currently understand the term "humanity"? Maybe not something as technical as Gwern's doom scenario, but still something that would be classified as pretty hard sci-fi and at the same time explore more the perspective of actual human characters on how living in such a world does feel like.
Even just very preliminary drafts would be very valuable I think.