Since the release of Android 6.0 Marshmallow, Gmail users have been reporting some syncing problems on their app that appeared simultaneously with their update to Android's latest flavor. (Very few users mentioned being affected on Lollipop, but it's worth pointing out.) The problems seemed to be divided in two categories: one that specifically stopped Hotmail accounts from being synced inside the Gmail app and another that caused a lot of delays in syncing and notifications regardless of the account type.
Gmail 5.10, which started being pushed to users on January 20, fixes the first issue. A Google employee explained on the corresponding Help Forums thread:
I can confirm that upgrading to the Gmail Android app 5.10 should fix this issue, however it's rolling out slowly so is only available for a small number of people right now.
I'll update this thread again once the roll-out is complete.
Thanks again for your patience!
However, the fix only works on the Hotmail account sync bug. If you're having random delays in syncing with other accounts or if sync has completely stopped working and is unreliable for you now, then you'll have to wait a bit longer for the solution. The team explains on the Help forums that it has identified the root cause of the issue but is still working on a fix:
The team has identified an issue which we believe is contributing to the syncing issue on Nexus devices. It is actively being worked on, and I'll continue to keep you all updated. In the meantime, please continue to post your experiences and/or if you have found any short-term workarounds.
One bug down, one more to go. If you still haven't received the update and can't wait to see your Hotmail account working again, Gmail 5.10 is available for download on APK Mirror.