Have called both companies and both are defensive and in 'not our problem' mode. So am trying to fix myself. Have checked Server Settings. Think OK. Incoming Port 993.. Outgoing Port 587. BT recommends IMAP with SSL enabled. And selecting 'My server requires authentification.' Only possible glitch I can see is that Outgoing mail server, in Apple's serving settings, appears to be 'offline'. Could this be the problem and how do I solve, please? A test of sending an email within BT's webmail interface works fine, so it probably is Apple's problem?
Try port 465 with SSL for Outgoing (SMTP) server, make sure authentication is 'Plain' (or whatever Apple's terminology equivalent is)
Try using a different email app such as Bluemail.
See link for correct settings.