There is obviously a bug in the log out section. If you can not log out you need to completely close your web browser by going into the Task Manager. To do that press the following keys... Ctrl , Alt, Delete all at the same time then click on the Task Manager or to take you directly to the Task Manger press Ctrl, Shift and Esc all at the same time.
When there you will see your web browser listed near to the top on the left hand side. Right click your mouse over it and you will get a drop down box. Select "End Task".
This will close your browser fully.
When you reopen it go to the BTMail log on page BTMail
If the "Keep me signed in" box is ticked try logging on and then log out to see if you are the required to log in again.
If it is unticked, tick the box then log back in and once logged in, log back out and then try logging in again to see if it now requires you to log in again.