The simple answer is don't do that! Modules calling each other or calling themselves are usually a sign of bad design.

Putting aside the validity of your design, let's first find the cause of the problem. Keep in mind that

use Module LIST;
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is the same as

require Module;
Module->import LIST if Module->can('import');
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while we look, line by line, at the code Perl executes:

test1.pl:2     use strict;
test1.pl:4     use lib qq{/usr/web/wonko/test/lib/SYS};
test1.pl:5     use lib qq{/usr/web/wonko/test/lib/AB};
test1.pl:7     require SYS_mail;

   Start executing SYS_mail.pm

SYS_mail.pm:1  package SYS_mail;
SYS_mail.pm:3  use lib qq{/usr/web/wonko/test/lib/AB};
SYS_mail.pm:5  require AB_prod;

   Start executing AB_prod.pm

AB_prod.pm:1   package AB_prod;
AB_prod.pm:3   use lib qq{/usr/web/wonko/test/lib/SYS};
AB_prod.pm:5   require SYS_mail;

   Does nothing. Already required

AB_prod.pm:5   SYS_mail->import(qw ( send_mail ))
                  if SYS_mail->can('import');

   ERROR!!
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The error is that we haven't yet executed

use vars qw(@ISA @EXPORT_OK);
use Exporter();
@ISA = qw(Exporter);
@EXPORT_OK = qw( send_notice );
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in SYS_mail.pm, so SYS_mail has no import function, so send_mail doesn't get imported into AB_prod. Similar errors occur at later points in the program.

And now the fix:

To fix this problem (aside from the obvious one of not including yourself directly or indirectly), change your modules to the following:

/usr/web/wonko/test/AB/AB_prod.pm
#--------------------------------------------------------------------
package AB_prod;

use lib qq{/usr/web/wonko/test/lib/SYS};

use strict;
use warnings;

BEGIN {
   our @ISA = qw( Exporter );
   our @EXPORT_OK = qw( send_notice );
   require Exporter;
}

use SYS_mail qw ( send_mail );

sub send_notice
{
   print qq{Hello from AB_prod::send_notice\n};

   send_mail();
}
1;
#--------------------------------------------------------------------
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/usr/web/wonko/test/lib/SYS/SYS_mail.pm
#--------------------------------------------------------------------
package SYS_mail;

use lib qq{/usr/web/wonko/test/lib/AB};

use strict;
use warnings;

BEGIN {
   our @ISA = qw( Exporter );
   our @EXPORT_OK = qw( send_mail );
   require Exporter;
}

use AB_prod qw( send_notice );

sub send_mail
{
   print qq{Hello from SYS_mail::send_mail\n};
}
1;
#--------------------------------------------------------------------
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Tested:

use strict;
use warnings;

use SYS_mail qw( send_mail );
use AB_prod qw( send_notice );

send_notice();

__END__
output
======
Hello from AB_prod::send_notice
Hello from SYS_mail::send_mail
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I also tested it with the two use statements inverted. It gave the same output.

Update: It doesn't matter that sub send_mail { ... } hasn't been reached at the point Exporter->import('send_mail') is called, so I removed the stub functions I added just below the BEGIN in SYS_mail. Same idea for AB_prod.