According to your issue, I guess you lost everything when you are updating to a newer iOS version. And I have tips below.
1. Make sure if you've backed up your data on iCloud before. You can login your iCloud account on the updated iPhone and check.
2. Check your computer and confirm if you've backed up your iPhone before the update.
If you have those update, congratulations, you didn't lose the notes at all. You can directly recover them from the backup.
If you haven't backed up those data, you can try to use some iOS data recovery tools to scan and check if you can get back the notes. And it's highly recommended that you scan as soon as possible since any further options and changes on the phone may rewrite the previous data. And if the data is rewrote, no one can get them back any more.
Here's a guide FYI. How to recover lost contacts, notes, reminders, and more?