I had same issue this week. BT said to remove and reinstall the app but cant do that on an iphone. Apple sorted it by deleting and re-adding the account. However now i have lost all my email folders on my iphone and also the ability to create new folders. Apple don't understand why this doesn't work (it used to!) and say it is an issue with BT. Nightmare...
I have had to delete my email account and re-add it on my iphone as i could not send emails due to unrecognisable password issues. Since then all of my email sub-folders (work, sport etc..) were removed but remain on the BT Mail app. I also cannot create new folders on my iphone wheras previously i could, Apple do not understand why this doesn't work and suggest it is a BT server issue.
I have moved you onto your own thread and I have merged your two posts because they are basically about the same problem and any replies you get will be relevant to your problem.
As regards you email account synchronising with the BT servers. This will only happen if you have set the email app up as an IMAP account. It sounds like you have set it up as a POP3 account.
Which email app are you using to access your email account?
Please keep all your posts on this thread.
Did you check to see if your email account is set up as a IMAP account or a POP3 account?
Are you using the correct settings as per these links?
If you are still looking for some assistance can you answer the questions I asked in message 6.