BA has launched an update to their mobile application today. It features a new digital version of your membership card:
The blocked out box has your membership number and name.
You can also secure details about all of your accounts, your Tier Points, Avios balance, and lifetime Tier Points also.
The Book with Avios option is right at the bottom of the page:
When booking with Avios, you can book wholly with Avios, or book and upgrade with Avios.
When you find flights you have to click to see which classes are available on your flight.
You can also select to part pay with Cash and use less Avios:
BA are moving towards one of the best apps in the industry with almost all of the web site features being available in the mobile app.