No probs here with my Touch ID. Tried to replicate it many times and nothing. Even waited a few minutes before trying again same. Rotation even works after unlock. Doesn't seem to be affecting everyone. This on my 6+ on 8.4 via OTA.
I had the problem last night where Touch ID would not register at all, not even if trying to set a new fingerprint. It didn't start working until I killed the Music app, but even then the login was definitely more leggy.
Fast forward 24 hours: it's all peachy. So here's my theory:
iOS 8.4, probably the music app specifically, does a ton of indexing when first installed. For those with a large music database on their phone (raises hand - 7,300 tracks) it perhaps had to do a bit of extra work to finish up. Now it's done, and it's all good.
Again, just a theory. We'll see if this continues.