# test e-mail june 2015
use strict;
use warnings;

use Net::SMTP;
use Authen::SASL;

my ($host_em, $to_em, $cc_em, $from_em, $subj_em, $data_em, %attach_em
+, $res_em, $err_em, $smtp);
my ($full_host_name, $password_em, $username);

my ($banner, $domain);

$host_em = 'mail.btinternet.com';      
                                                             
$full_host_name = (gethostbyname $host_em)[0];                  
print "full_host_name <$full_host_name>\n";                     

$from_em   = 'valid e-mail address'; 

$cc_em   = 'valid e-mail address'; 

$to_em = 'valid e-mail address';

$password_em = 'password for $from_em';

$subj_em = 'Test e-mail';

$data_em = 'Please let me know if this arrives';

$smtp = Net::SMTP->new($host_em,
            Debug => 1);

print "\nsmtp <$smtp>\n";

$smtp->auth ($from_em, $password_em);

print "\nbefore ->mail\n";

$smtp->mail ($from_em);

print "\nbefore ->to\n";

$smtp->to($to_em);

print "\nbefore ->data\n";
$smtp->data;

$smtp->datasend("From: " . $from_em);
$smtp->datasend("To: " . $to_em);
$smtp->datasend('Subject: This is a test');
$smtp->datasend("\n");

$smtp->datasend("$data_em");

$smtp->dataend;
$smtp->quit;

$banner = $smtp->banner;

print "\nbanner <$banner>\n";

$domain = $smtp->domain;

print "\ndomain <$domain>\n";
[download]

Can anyone suggest how I can make these e-mails actually arrive and/or find out what is happening to them?

Contact the administrator

When a SMPT server tells you, you've authenticated, your mail has been sent ... your only choice is to believe it

I can see your very reasonable point?
I see you suggest I talk to the administrator but I am not sure who this is this particular case.
I get my broadband services from Virgin Media but use an e-mail account from bt (British Telecoms). Therefore does this make it bt that I should talk to? Using Thunderbird with the from address I am using does work OK.