This issue was driving me crazy so I decided to restore my phone set up brand new. Set up on 8.4 as a new phone without adding any extra apps or even signing into iCloud. In fact, the only apps I ever opened since the restore are Settings and the Camera. Before I tried out 60fps even once, I made the following settings...
. turn off Assistive Touch
. changed view from Standard to Zoomed
. disabled rotation
...and then powered down the phone, waited a minute, then powered back on. Double checked the changed settings remained in tact, then closed (swipe-up closed) the settings app so no other app at all was running.
All the above to say: I took every possible measure to get the 60fps video to work. And when I opened the camera app and swiped to video, it still froze up.
While some of the fixes in this thread may help you, it's very possible that none of them will. This is a bug that, in my case, fully negates one of the major selling points of the phone.