Ok I registered on this website just to post what worked for me because absolutely none of these worked and I found the actual solution.
Samsung has been scared of the battery fiasco since the note 7 and since the battery in the s8 is smaller they've basically placed Gmail and all other apps into a power saving mode. I was having push notification problems with Hangouts too by the way.
1. Go to phone settings.
2. Click "Apps"
3. Press the 3 dots at top right corner and press "Special Access"
4. Press "Optimize Battery Usage"
5. At the top left it says "apps not optimized", press this until it says "All Apps"
6. Scroll down to Gmail. You'll notice that it is "on" with a blue indicator on the right. You actually need to turn this "OFF", so that Gmail goes into the "apps not optimized" section.
This is the only thing that worked for me after clearing cache, reinstalling, and all the other stuff posted.
I hope this helps. Nougat 7 settings and Samsung s8.