Thank you for the suggestion. I tried this this morning but it has made no difference - I still had to delete all browsing history, cookies, etc. before being able to log in.
The other very annoying aspect of this issue is that BT has the habit of logging me out of email (when I am, for example, working on a document or spreadsheet), which then means that I can only log back in after going through the clearing browsing history, etc. process.
After you have cleared your browser's history and cookies, the next time when you log onto your BT account you should be offered the cookies again. Do not accept them, reject all and if you are on any other BT site that offers cookies reject them as well and see if that helps.