Sounds like Sketch would be the best way to tackle this, since you'll be needing its tools for layer styles (editing one affects all others), text styles, and batch exporting to multiple resolutions. Check out this Sketch 3 intro document.

Compared to Photoshop and Illustrator, I think Sketch strikes a good balance between shape editing, layer styles, artboards, and an emphasis on pixel-based work. Trying to create an app of this size in either Adobe software would become unmanageable very quickly.

For my most recent Sketch project, which also consisted of many app screens and reusable UI components, I structured my workflow like this:

• Draw an artboard for each individual view/screen in the application. Create layer styles for reusable UI components as you go so that changes to one UI component will affect them everywhere. I arrange my artboards roughly in the pattern of a sitemap or user flow, and they're easy to rearrange as necessary.
• Attempt to illustrate screen views that demonstrate the maximal set of elements for that screen, and draw any alternate states for
• Then, create a Page 2 in the Sketch file for individual assets, like backgrounds and icons. I copy each element from its screen view, and paste it into its own artboard on Page 2.
• Finally, I select each artboard on the layer menu (for both pages) and give it some Export resolutions before batch exporting. For web, this would usually be @1x and @2x resolutions.