Is there a way to (or another app that) automatic download new tweets?
The iOS Mail app has special abilities not available to 3rd party apps, under the App Store rules. Specifically it has greater leeway on how it can act in the background. 3rd party apps can't update every x minutes, nor can they receive a trigger when connectivity is restored (I'm not actually sure that Mail.app gets this either, it may just be that it's push/scheduled check doesn't fail once connectivity is restored), although they can trigger downloads based on geofencing.

This sort of thing would be possible on a jailbroken device, but I don't know if any apps actually offer this functionality, perhaps someone can chime in.