That issue has happened to me previously.
As a work around until it was resolved I did as you are doing and that was to clear history and cookies (from all time) then used the direct logon page ie click on the email button at the top of this page. When the logon page opened I then saved the opened email page as a "favourite" .
Once the logon was established I then logged off and closed the browser. I reopened the browser and using the "favourite" page which now brought up the logon page and I was able to log on again repeatedly without issue and without clearing the history etc.
I never reported a fault with BT because the problem resolved itself at some point and I could logon without the use of the "favourite" logon page. I do not know how it was fixed or if it was just one of the BT mail servers playing up.
If it is an email server playing up you could try and switch off your BT hub for about 30 minutes and when you turn it back on you should have been issued with a new IP address and hopefully connected to a different email server.
If that does not work for you in order to get a permanent fix you will need to contact BT 0330.1234.150 and ask for the email team and explain the problem including what happens when you do try to log on as per your last post.
