I've used bt mail for many years, and seldom had a problem with it, whether going in from OE in the past, or outlook now, or going in via their website. The website allows access from any of the gadgets around the house - Apple, MS, old, new etc.
So I cannot add to the technical side of the question.
There is a little trick to check. On the lappie, are you going in via the BT.COM website? There used to be another access point, which took you through a more Yahoo-ey front end. This no longer works. Confusingly, it looks similar to the BT.com entry.
Now you need to go to BT.com, then click on Existing Customers, then click on e-mail, then sign in to e-mail.
Another little check is that my BT com access from my Android does not allow me to save sign in details - or, rather, it tries to save them, but BT responds with the password not recognised reply. So I type in both my email address and my password each time I access the email via android.
I will, however, endorse what mixture says - get serviced by that Chairman/CEO office at BT, and things happen.