BT Internet uses the defunct SMTPS protocol for outgoing SMTP connections from a customer.  To set up Exim4 to be able to send mail via this interface requires a dirty hack.  The main part of the configuration is as described in this Ask Ubuntu article in the accepted answer (based on Tony Scelfo’s article).  However, that does not take account of BT’s peculiar configuration.  An additional step is required:  before running update-exim4.conf, edit /etc/exim4/exim4.conf.template. Find the section starting with

remote_smtp_smarthost:

Insert a new line following the line driver = smtp like this:

  protocol = smtps

Then continue with the steps in the article above. Use mail.btinternet.com::465 for the address of the smarthost when running dpkg-reconfigure.

This is a hack because it subverts the SMTP smarthost function to use SMTPS. However as you can only have one smarthost at a time, that doesn’t matter.

By the way, Raspbian on the Raspberry Pi uses Exim4 and this was how I got email working from it.