Had major issues on two separate iPhones after upgrade to iOS 7. Emails were often rejected on personal and office accounts (latter through MobileIron) and when they were sent, there was a delay of 10-90+ minutes with no reliability.
Tried many of the above solutions including lengthy visit to the Genius Bar and conversations with cellular providers. Ultimately, was persuaded that the only solution was to Restore as a New iPhone......no mean feat given over 800 apps!!
Undertook major rebuild of both iPhones and documented below for completeness:
1. Restored from new and tested emails before installing apps - immediate send/receive
2. Rebuilt one iPhone with installation of 600+ apps including configuration into folders - huge task!
3. Tested emails again - all functional on fully new iPhone
4. Wiped and then restored iPhone 2 from new backup
5. Tested: all fine with immediate send/receive
Interim steps included adding MobileIron, only relevant to those specific users.
Looks as if problem has been resolved - presume some form of corrupt file on two iPhones from two original and separate builds/backups. Unclear whether iOS 7 caused the corruptions but hope the above will save people the days of anguish which I suffered!