I have BT mail set up on an ipad, my desktop pc and on an Apple phone. The ipad and pc both seem to work correctly my Apple phone receives and sends mail ok. if I try to delete mail on this phone, it is erased from my iphone but the incoming mail counter still shows the mail that I have opened and deleted in its count. my pc and my ipad both seem to work correctly and show the same amount of mail which is lower than the hundreds shown on my phone, i suspect that my phone still thinks the old emails are in it's list and does not delete them from my other devices.
What email app do you use on:
Have you checked to see if your iPhone email account is set up as an IMAP account or a POP3 account?
I am unsure how i find out, in the iphone settings for my mail account under the heading outgoing mail server it refers to SMPT and shows a server port 465 (which I believe is correct for SMPT), underneath that it lists an Advanced heading, where it lists a server Port 993 which I believe is correct to IMAP,
I am reluctant to just delete the account from the phone, as I had a look on the App store and I believe that I would end up with an additional app if I try to install BT mail from there.
Going by the settings you have posted it is set up as an IMAP account.
If you want to delete an email account from an email app you don not need to uninstall the email app to achieve that and as such you do not need to re-install the email app. You would only need to set the email account up again in the app in same way as you did when you first set it up.
Is it the BTMail app that you are using?
If it is, try setting up the Mail app that was pre-installed on the phone to see if that works. If it does you can then either just stop using the BTMail app or uninstall it.