So far, the app has appeared in Apple TV search in the US and Canada, but not the UK, according to my Engadget colleagues there. The best way to find it, notes 9to5 Mac, is to search Amazon in the tvOS store, select Amazon Shopping and scroll down in the "more by this developer" section. As you'd expect, it's much like the web, Roku and smart TV apps, letting you scroll and search films and TV series, select favorites, and more.
Amazon Prime Video was supposed to come to Apple TV by this summer, but was delayed for unknown reasons. The service, which is available to all Amazon Prime members, was already long overdue when Cook announced it was coming in May of 2017.
Google recently pulled YouTube from FireTV and other services, but Amazon has also feuded with Apple in the past. The two companies famously fought over e-books and commissions from items purchased using Amazon's iOS app, and (other than briefly), Amazon hasn't sold the Apple TV since 2015. Given the benefit to both Prime Video and Apple TV users, it's good to see they can agree on at least one thing.