If the emails are not on the BT server when you log onto your email account using a browser it means that yes they can not be downloaded but it also means that if they are not on the server they are lost and can not be recovered.
As regards why the emails are missing there could be a number of reasons such as an email client\ email app or the BT servers triggering the deletion or user error deleted them.
If user error triggered the deletion the deleted emails would normally be in your Trash folder and I am assuming that you have checked there and they are not in the Trash folder so that would in my opinion rule that out as being the cause.
You will need to contact BT 0330.1234 .50 and ask to speak to the email team and request that they investigate why the emails are missing and if they can be recovered.
Having said that, unfortunately due to similar previous posts on the forum with the same issue, when investigated the email team said that the emails could not be recovered and that all they can do is inform you which device triggered the deletion.
Not much use now but in future I would suggest that you do not use the Inbox as a storage folder. You should set up named folders and store your emails there. If you are using IMAP these folders will be replicated and synchronised on all your devices and be accessible on those devices.
