Ok the simple solves would be to uninstall/install the app again.. I would also download a 3rd part app and check if I had the same issues with that one; if the issue is still there in that case then it's most likely due to twitter
I just fixed it by following the advice on a previous post - go to the twitter website in your browser, go to settings/apps and revoke access to the Twitter for Windows Phone app. Uninstall / re-install the app from WP and voila, all notifications work normally! I had granted access to the app waaay back in 2012 and Twitter probably did some server side updates that was causing notifications to fail for older apps. Renewing the app in this manner fixed the problem - go give it a try!