What seems to be happening is that, when sending the read receipt it is using an address other than yours for some reason.
When sending mail through the ntlworld mail server it checks the sending address against the list of known email addresses and if it does not match or is blank, you get 500 relaying denied
unless you are sending to a valid ntlworld.com email address (which is why you could send the read receipt back to yourself) as you can see below (I've masked my own email address)
220 ESMTP server ready
helo ntlworld.com
250 aamta05-winn.mailhost.ntl.com
mail from:<
****@ntlworld.com>
250 Sender <****
@ntlworld.com> Ok
rcpt to:<
test@imperial.ac.uk>
250 Recipient <
test@imperial.ac.uk> Ok
rset
250 Ok resetting state
mail from:<>
250 Sender <> Ok
rcpt to:<
test@imperial.ac.uk>
550 relaying mail to imperial.ac.uk is not allowed
rcpt to:<
****@ntlworld.com>
250 Recipient <
****@ntlworld.com> Ok