iPhone 6S - Not getting Push Notification and Badge icon related to OS Updates

The problem with this type of question is that it can only be verified as resolved under the right circumstances.

Although you say you have reset the iPhone three times, and once at an Apple service centre, it's not clear what type of reset was done. My advice is to follow these steps to try and resolve your issue.

1. Connect your iPhone to your computer and launch iTunes (if it isn't already open). Note: You may need to enter a passcode or trust this computer.
2. Select your iPhone when it appears in iTunes. (Note: If you're prompted to update iOS do not download/install the update.
3. Manually do a full backup of your iPhone to your computer (i.e. not just to iCloud). You can do this by clicking on the Back Up Now button. This may take some time depending on how much data you have.
4. Once your backup is complete, disconnect the iPhone from your computer.
5. On the iPhone go to Settings > General
6. Swipe up and tap on the Reset option at the bottom
7. Tap on Erase all Content and Settings
8. Follow the prompts to erase/reset your iPhone
9. Once you've reset your iPhone, reconnect it to your Mac or PC and launch iTunes (if it isn't already open). Note: You may need to enter a passcode or trust this computer.
10. Select your iPhone when it appears in iTunes
11. When prompted, choose to Restore your iPhone from the backup you made at Step 3
12. Follow the prompts to restore your iPhone

Once these steps are complete you will have hopefully resolved your issue.

At the time of writing this, iOS 10.2.1 was just released (hence why at Step 2 I didn't want you downloading/installing the update if prompted).

If you haven't already installed this update, then you should get your notification soon. On the other hand, if iOS 10.2.1 was already installed then you won't know if these steps have resolved your issue until Apple releases the next update.

EDIT re: Timing

I just wanted to add a bit of info on the timing of notifications relating to iOS updates.

Typically, after an iOS update is release, users should get a badge notification with a '1' that sometime in the first 48hrs - 72hrs. However, an actual push notification may not come until quite some time after and only if your device hasn't been updated yet.

Using myself as an example, I have an iPhone 6s which I've had since September 2015 (it's now Feb 2017). In that whole time I've never had a push notification about an iOS update because I always update my iPhone fairly quickly. However, my wife who also got an iPhone 6s at the same time as I did, did get a push notification for iOS 10 at some point (weeks after iOS 10 launched) because she was still on iOS 9.3.5 and was in no hurry to update (I'm usually her guinea pig with these things). However, she did eventually update from iOS 9.3.5 to iOS 10 because the notification was relentless annoying her (and because iOS 10 was working well for me).

In summary, what I am saying is that a badge icon should appear within 72hrs of the official release, but that a push notification message may take weeks and only if arrive if you haven't updated.