Q: Apps won't update or download from App Store

What have you tried so far ? Have you tried :

- logging out and back into your account on the phone ?

- disconnected and reconnected to your network ?

- closed the App Store app via the phone's taskbar ?

- a soft-reset/reboot of the phone ?

And what else have you tried ?

Try a different network. Or you could try downloading one of them on your computer's iTunes and sync it from there to your phone and then see if other updates and/or new apps download directly on your phone.

You have a recent OS on your computer, and iTunes 12.4.3 installed on it ?

Device not showing in iTunes : If iTunes doesn't recognize your iPhone, iPad, or iPod - Apple Support