I have the same problem. We have two email addresses in the household. When I select logout from the drop down window, it does not show the login screen, but reloads my email again. I am using Windows 10. But It only happens on Google Chrome, which is what I mainly use on my PC. I do not have the problem on Microsoft Edge. So for me it seems to be a problem with BT Email support for Google Chrome.
I have the same problem, I'm on Windows 10, using the latest Google Chrome, went just now to check my mothers email box and can't log out of my email account. BT email just refreshes and I'm straight back in! I will look at the cookies etc, but how ridiculous, this has never happened before, what have you done BT?