Hi! I'm having the exact same problem! In addition to not being able to take any new photos (I click shutter button and nothing happens - It remains opaque as if I never touched it,) I can't delete any photos, nothing saves to photo album, AND I cannot do the update!
I have an iPhone 5S and it seemed to start all of this after I cancelled the iCloud extra storage charge since I always had 49 GB left & I never used it.
Has anyone had ANY luck on fixing it or figuring out what's causing this??
If this is an attempt at forcing us to upgrade, I'll just switch to the other side and go with a Galaxy..it's ridiculous to think but oh so probable.
/rant