Updating apps doesn't work for me either, on two devices. I can see the updates but when trying to download them it just keeps spinning. Not sure if it's iOS 8.4 or some problem in the app store (Apple's status page doesn't show any errors though).
OK, today I still had the problem. I then killed the App Store app in the task switcher, logged out of the app store in the settings, and rebooted the phone. Then logged back in, opened App Store, and now the updating works. Yet another 8.4 glitch, I guess.
Any you asking for the world. To think that you want to get Apple to actually put out an update that doesn't break more than it fixes.
BTW: the issue is being caused by the update to iTunes Store agreement that you have to accept, but sadly the dialog box didn't come up on your screen. You can try to log out of your AppleID for the store, then log back in. This might get the agreement to pop up. If not, then you need to restore from a backup (I assume you made one on iTunes on your mac) before doing the upgrade to 8.4.