Hmm, have you tried the following steps below? Try it with the official app... but when you uninstall and reinstall the app, do a reboot and you should be ok
Feel free to let us know if it worked out for you, or if it didn't work.
Really? Still nothing? Hmm. Is twitter updated? I suspect a problem on twitters side, possibly a bug within the app. This has been reported by many, some can get working and some can't with the same methods. Some even say push notifications on the official twitter app never worked for them, but I suspect that it has once worked for you, right?
Do you know anyone else with this problem?
This was posted here at Windows Central a while back
The official Twitter app notifications aren't reliable at all. There's been times where I've checked and has tons of notifications I never knew because they never went off