Unfortunately it seems that there is a chance it could be delayed. In a report from Recode, it seems that Amazon Prime Video might not be ready just yet. Apple is rumored to be launching a new Apple TV with support for 4K alongside the new iPhones and Apple Watch this coming September, and that the launch of Amazon Prime Video could also be taking place then.
However the report from Recode suggests that it might not be launched in time, although it does not mention specifically as to what could be the reason. However the report does cite roadblocks that Apple is facing with its Apple TV plans which is similar to what we have heard as well, although it is unclear if this has anything to do with the launch of Amazon Prime Video especially since other video platforms already exist for the Apple TV.
In any case we will have to wait and see if the launch of the app has really been delayed, so until then do take it with a grain of salt.