I don't use Apple devices but I suspect the 'Password' option is plain password authentication.
I don't think that is your problem, if it was there would be thousands of complaints on the forum.
I assume you are either a BT broadband customer or paying for Premium Mail if not.
As far as I am aware the Authentication is "Password".
I also do not use an iPhone so can not check to see what setting is used but you must be doing something different to the thousands if not millions of BTMail customers who use BTMail with the settings provided by BT on their iPhones and other Apple devices as no body else has mention the problem that you have.
Have you tried deleting the app and then re installing it to see if that helps.
Perhaps a forum member that use an iPhone can post what their authentication settings are.
I presume you have read message two. Can you post your answers to the questions asked.
Have you checked in your MyBT > Your Products > Email, manage to see what type of email account you have. Is it Basic, Standard or Premium?
> Premium
Has your email account worked previously on your iPhone?
> It's an iPad, but yes, for several years - and no changes had been done to the Mail settings.
Does your email work when using a web browser (web mail)?
> Yes
Which email app are you using?
> iOS Mail (Apple default app for iPad)
Are you trying to set up an IMAP account or a POP3 account?
> IMAP
What has caused you to have to set it up now?
As per message 7.
I would delete the email account and to be sure of clearing out any cached settings you should restart the iPad and start again following what I posted in message 7.
It may not be necessary to change the password if you do not want to unless you have already changed it in which case I would change it again and follow message 7
Prior to doing that check to see if there any updates for the email app that you are using.
If it still does not work, for test purposes you could try and set up the BTMail app to see if that works.