Using LWP::UserAgent (which will follow redirects) I got it to work:

use LWP::UserAgent;
my $url = 'http://www.bbc.co.uk/radio4/programmes/schedules/fm/2015/10
+/13';
my $ua = LWP::UserAgent->new(ssl_opts => {verify_hostname => 0});
my $res = $ua->get( $url );
my $html = $res->content;
print $html;
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