Found this explanation of the problem:
At last I have a solution following several online sessions with Yahoo help desk operators and phone calls from one of their Supervisors.
Summary of the problem: Yahoo, which is used by btinternet or BT Mail say that they have recently introduced changes to their system to no longer allow two devices to log onto and collect mail at the same time from their server. They say that this is to improve e-mail security and is for our benefit.
So, if your iPhone or iPad is set to automatically fetch mail and you are also connected to the same mailbox on a Mac or PC, the desktop application will at some stage be locked out by Yahoo and the indication of this is the MacBook or PC asking for your user name and password. When you input the username and password it will be rejected. On the face of it, this is a bit odd and suggests that you have the wrong password or user name or errors in your settings and you probably have these all set correctly.
If you then go to web mail directly at Yahoo.com, then log out of your account and back in again , the process will reset the system allowing you back into your desktop mail system. This solves the problem for a short time until the iPhone or iPad fetches mail again automatically and you will find yourself in a frustrating loop between the desktop mail and web mail. This will be even more maddening if you have more than one mailbox on Mac Mail.
The Solution: Finally, after a week or more back and forth with Yahoo, I was advised to turn off the automatic downloading of e-mail on all my devices and then to fetch mail manually and this seems to work fine. 24hours later I have not had further problems.
On the iPhone and my wife's iPad this means that mail is collected only when we click the mail icon and so there are no longer any audible or visible prompts that new mail has arrived. This is not ideal and does not seem like the progress that Yahoo claim for my e-mail experience but may be better than moving to another provider involving changes of e-mail address.
Its just a pity that Yahoo or BT did not warn us about their new security plans or provide guidance about managing multiple devices going forward. But frankly, I seems likely that the consequences of the changes that they made came as much of a surprise to BT and Yahoo as it did to us.