Did you confirm the Fetch settings on the phone? It should default to 15 minutes. Settings -> Mail -> Accounts -> one of the accounts. When you open the app, it does an immediate fetch, so if you don't let it go 15 minutes at least, you won't get a new mail indication, which is why it might look like it's not working.
I've never heard of settings in Dovecot / Exim - maybe you can link them?
