Presumably you are searching when you log onto your email account via a browser and not searching on an email client/app.
If the emails are not showing up on the server it means that they must have been deleted at some point.
Assuming you are now on the BTMail email system and not the BTYahoo email system, going by the date 2014 it may have coincided when/if your email account was migrated from BTYahoo to BTMail which was taking place in 2014.
It could also be the case that if you were using a mobile device prior to 2014 that you had the email app on that device set up as an IMAP account which would mean if you deleted emails on the mobile device it would have deleted them on the BT server or if you had the device set up with as a POP3 account and had not ticked the box "leave a copy on the server" it would mean that the emails would be deleted from the BT servers after you had downloaded them to the email app.
In any event regardless of how it happened if the email are not on the server they will be gone forever now.
