I was seeing this last week, then magically it cured itself. If I recall, the issue was only on 3G/LTE and not while using Wi-Fi. So either wait to see if it gets better, or try updating on Wi-Fi, or update all your pending apps from your computer and re-sync your iPhone as suggested by Randers, or try Resetting or Restoring your iPhone (worse case).
I rebooted my computer and now it's working.
When in doubt, reboot, but in my frustration this afternoon it didn't occur to me.
Thank you all so much for taking the time to answer my question.
I hope this simple solution will help someone else.
I get frustrated with Apple too, but if you've spent any time on an Android device, you would know how lucky we are to have an Apple product. The interface is awful and there are no apps, even to get stuck.