The vast majority of btinternet.com email addresses are - and always have been - with Yahoo. Some, however, are with another email provider. The transfer from Yahoo to the other provider did not go well so further transfers were stopped at least 2, possibly 3, years ago.
Anyone logging into their BT Yahoo mail via the bt.com website recently will have seen warnings to change their passwords and other security information. Emails were sent to all BT Yahoo customers when the Yahoo breach was disclosed several months ago. I bet the majority of recipients ignored the advice though!
The problem with some email providers rejecting email as spam is usually caused by "SPF" (Sender Policy Framework).
SPF uses the "reputation" of the sender's email provider's IP address to determine whether or not an email is spam. As thousands of email users will share the same IP address it only takes one email user who sends out spam to tarnish the IP address reputation. Thus, for a short while, emails might be rejected as spam during the period that the IP address is blacklisted. Lots about it on the web if you want to search for it.