Thanks to everyone who answered privately.
In summary, BT has been moving users off Yahoo and back onto their own
systems (*).
If when you login to read mail online you get to
https://btmail.bt.com/cp/ps/main/index#mail
(which I did) then this is a migrated account.
This means you have to connect using manually-configured IMAP rather
than the Yahoo! account settings in iPad.
Cheers,
Brian.
(*) Some fluffy information at
http://bt.custhelp.com/app/answers/detail/a_id/44789/~/we%27re-changing-bt%27s-portal-and-email-services
