EDIT: This post crossed with your above replies so some of it may not be relevant.
The family members that have an email address from you MUST have the email account "gifted" to them and they MUST set up their own BTID so that they can manage the email account and if need be set up 2FA.
If they are not "gifted" the email account and then set up their own BTID their email account will be seen as one of your email accounts and any 2FA set up will notify you and you will have to acknowledge it every time 2FA is activated.
If you "gift" the email account to them and they set up 2FA it will notify them, assuming the have have a mobile phone, and they will have to acknowledge it in order to access then email account.
See link about "gifting" email accounts.
How do I create and activate a new BT Email address? | BT Help
As regards your own 2FA set up, are you the broadband account holder and is the email account one that you set up on that BT Broadband account or is it one that was set up on a previous BT Broadband account such as a parents account or a previous BT Broadband account that you may have had?
If it was not set up on the present BT Broadband account you will need to associate it to that account so that you can manage it and set up 2FA.
See link
Can I move my BT email address to another account? | BT Help
