I would not get your hopes up about compensation for your email not working on an email client/app.
BT supply email as an "add on" which is delivered by webmail via a web browser. Even although email clients/app can be used and BT give the relevant server details in order to use them, they are not supported by BT with the exception of the BTMail app.
As you stated you were getting your email when you logged on via a browser it would appear that your email service was working albeit as a "Basic" account not a "Standard" account.
If you do decide to try and get compensation please keep this thread updated with how you get on.
This problem is now resolved. I don't know what they did but someone at the email Hub phoned to say that they had identified a mismatch in their settings.
If anyone reading this thread has this problem I suggest you phone customer service, insist that they escalate your complaint and refuse to close the complaint until it is resolved.
And when they tell you that someone will phone you within a few hours.....expect it to be considerably longer. I finally got a call from BT 3 days after the stated time. Some people might say that I DID get a call. And that is true. But I am paying for this service, I think we should all expect better. And if BT sorted out their processes and procedures perhaps they wouldn't be so overwhelmed with issues that, according to them, 'shouldn't have happened'.