I believe you'll need to turn on mobile notifications for each contact. Instead of sending text messages, since you have the application, it'll send push notifications which will show up in NC.
Downside: you need to remember to do this for each person you follow, which is a pain, and you need to remember to do it for each new person from here on out. I'm not positive, but this might need to be done via the website.
Upside: you could have only some people's tweets get pushed to your phone if you have some people who are annoying, but for whatever reason, you don't want to unfollow.