Since no one from apple has provided a helpful suggestion, I'll venture this one.
Unfortunately, some updates force apps to break just a little when they have to adjust some security settings.
What I have tried (just today) is:
- Go to Settings > Mail > Accounts
- your accoutns will be listed there.
- Tap one by one, and check for those that say in red letters "Re-enter password"
- Do not re-enter your password (as we have seen, this doesn work).
- Instead, go down to delete account, confirm "Delete from my iPhone"
- Then go back to accounts, and add the account back in by tapping "Add Account"
- Then when the password is asked, enter the password manually, do not use the iOS keychain saved passwords. (this could also be broke or corrupted in some way after the upgrade, better to be safe)
- When you enter the password manually, keychain will also update itself with whatever you just entered as a password.
I still do not have evidence that this actually works, but being in IT and security support for a while, leads me to think that is the best route to go... considering that apple may never even see or reply to these posts.
If you think this helps, please mark it as helpful and share your results with a reply so other may benefit from this, or flat out know that my suggestion does not work at all :P (please be constructive with your feedback)