It would depend on how you set up your Apple Mail app whether or not that your sent emails would show up on BT webmail.
If the Apple Mail app is set up as an IMAP account it would show because IMAP accounts are synchronised with the BT webmail servers. ie what you do on one would show the same on the other such as delete or send an email on webmail it will show on Apple Mail and visa versa or set up a folder on one it will show the folder on the other.
If it is set up as a POP3 account it will not be synchronised. It will show what you have done on Apple mail but it is not synchronised/replicated on BT webmail. ie a shown sent or deleted email will show on Apple Mail app but not on BT webmail.
You can check to see how it is set up by going into the Apple Mail app settings.
If the email shows as sent it most likely that it has been sent. If it was not delivered, in most cases you would receive a bounce back notification but that is not always the case.
The only way that you will know for sure if the company received your email would be to call them and ask or wait until the closure date for the job vacancy has passed by which time if you have not been asked for an interview it is likely that you were unsuccessful. Unfortunately few companies bother to inform people that they were unsuccessful.
