How are you accessing your emails on the iPhone and Mac?
Are you presently a BT Broadband customer or are you paying for your email through a BT Premium email account or did you migrate to EE Broadband?
If none of the above your email account may have been downgraded to a BTMail Basic account which can not be used with an email client or email apps.
See link
https://www.bt.com/help/email/bt-email-products
If you are a BT Broadband customer and you are are using an email client to access your emails on the Apple Mac it may be that you have it set up as a POP3 account and the box to "leave emails on server" in the settings has not been ticked.
This means that any emails that the Mac receives will be deleted from the BT server after it has downloaded them and as such will not be available for your iPhone to access.
If that is the case you will need to go into the settings and tick the box.
