Q: iPhone won't connect when not on wifi

Try these steps If you can't connect to a cellular network or cellular data - Apple Support

If you're using cellular service, make sure that cellular data is turned on from Settings > Cellular. If you're connected to cellular data, larger items might not download. You might need to connect to Wi-Fi to download apps, videos, and podcasts.

If you can’t connect to the iTunes Store

Please do not start duplicate discussions ...

iPhone won't connect to App store

It makes it that much harder to assist you.

Go to:

Settings > iTunes & App Stores

and make sure "Use Cellular Data" (near the bottom) is not turned off.