This started about a week to ten days ago for me (now 20-04-24). I have my email account open all the time at home as I am constantly using it and receiving replies. Now it just flips off closed every so often. It is driving me mad having to keep logging in, and then when I do it comes up with another BT home page and I have to click on 'email'. There doesn't seem to be a "keep me logged in to emails" option any more.
The option has been removed since the latest update to BT webmail.
Use an email client/app instead of webmail. What device are you using?