BT Email hacked and now cannot get Apple devices to accept changed password

Have you tried removing the account from your devices and re-adding it? If so and that's still not working, you will probably need to address this with BT support. As has been the case with other email providers, they may now have additional security procedures when connecting an email application to your email account.

Regards.

Can I ask what the solution was in the end?

I have a friend with exactly the same issue and would love to help him.

I've been at it for an hour and am making no progress - do I need to call BT support to get it solved?

What is a BT email? That said it sounds like whoever hacked your account has changed your Apple account password.

Glad you were able to get it sorted. Cheers.