Sometimes apps can be side-loaded on the Fire devices, as it's Amazon's version of Android. And I think there are screen mirroring apps that might work from the iPhone to the Roku but they can be iffy with paid streaming services. (Not an Apple person, so that's the extent of my knowledge.) The computer might be the only option at first.
I forgot to add that there will be an app for Chromecast and Android TV, too. We have an older Shield in the bedroom, so I may watch on that. It does seem odd that Roku won't be available on launch but I suspect that Roku and Amazon (FireTV) may want more than NBCU is willing to give at the moment. Who knows, maybe there will be a late-breaking deal.