iPhone doesn't automatically get inbox' child folder mail
You need to pick folders to push if you want to sync certain sub-folders for notifications. To do this, go to Settings > Email > Accounts > Fetch new data > iCloud, and choose which folders should be pushed automatically.

I'm using a free app called 'myMail' that lets me choose the sub-folder to get the notification when there is new e-mail. It's working perfectly with my Gmail, Hotmail and IMAP account that has around 10 sub-folders. You can try it too.