Hello ishanchatterji,
Welcome to Apple Support Communities.
We understand that the Notification Center on your iPad doesn't appear as expected until you've opened it completely.
In such a situation, restarting the device is a good first step (Restart your iPad).
If the Notification Center still fails to appear as you'd expect, let's make sure that you've a current backup of your iPad (Back up iPad).
Next, reinstall iOS on your iPad by following the directions outlined below:
"1. Make sure that your Mac is up to date. If you're using iTunes, make sure you have the latest version of iTunes.
2. On a Mac with macOS Catalina 10.15, open Finder. On a Mac with macOS Mojave 10.14 or earlier, or on a PC, open iTunes. If iTunes is already open, close it, then open it again.
3. Keep your device connected and wait until the recovery mode screen appears:
- iPad models with Face ID: Press and quickly release the Volume Up button. Press and quickly release the Volume Down button. Press and hold the Top button until your device begins to restart. Continue holding the Top button until your device goes into recovery mode.
- iPhone 8 or later: Press and quickly release the Volume Up button. Press and quickly release the Volume Down button. Then, press and hold the Side button until you see the recovery mode screen.
- iPhone 7, iPhone 7 Plus, and iPod touch (7th generation): Press and hold the Top (or Side) and Volume Down buttons at the same time. Keep holding them until you see the recovery mode screen.
- iPad with Home button, iPhone 6s or earlier, and iPod touch (6th generation) or earlier: Press and hold both the Home and the Top (or Side) buttons at the same time. Keep holding them until you see the recovery mode screen.
4. Locate your device on your computer. When you see the option to Restore or Update, choose Update. Your computer will try to reinstall the software without erasing your data. Wait while your computer downloads the software for your device. If the download takes more than 15 minutes and your device exits the recovery mode screen, let the download finish, then repeat step 3.
5. After the Update or Restore completes, set up your device."
If you can't update or restore your iPhone, iPad, or iPod touch
Hopefully this helps!