There is no firm date. the migration has taken longer than expected. You will just need to wait until it happens.
You will NOT receive an email telling you that you are being migrated and asking you to verify or supply details.
You will receive one after you have been migrated informing you of that you have been migrated with links about what you can do with the new system and help links. There are no links asking you to supply details or to log onto BTMail. If the email has any it is a phishing email.
There are a few changes that you will be advised of the first time you log onto your email via webmail.