Why is wp_new_user_notification not working?

Try this code. I have removed the new declaration of wp_new_user_notification function from the code as it would consider the new function only if it is put in a plugin. That’s how the priority is taken.

<?php
/* Check if users can register. */
$registration = get_option( 'users_can_register' );

/* If user registered, input info. */
if ( 'POST' == $_SERVER['REQUEST_METHOD'] && !empty( $_POST['action'] ) && $_POST['action'] == 'adduser' ) {
    $user_pass = wp_generate_password();
    $userdata = array(
        'user_pass' => $user_pass,
        'user_login' => esc_attr( $_POST['user_name'] ),
        'first_name' => esc_attr( $_POST['first_name'] ),
        'last_name' => esc_attr( $_POST['last_name'] ),
        'user_email' => esc_attr( $_POST['email'] ),
    );

    function make_blog_name_from_name($name) {
            return 'RNA Administrator';
            }
            add_filter('wp_mail_from_name', 'make_blog_name_from_name');

    function from_mail($content_type){
                return '[email protected]';
            }
            add_filter('wp_mail_from','from_mail');

    if ( !$userdata['user_login'] )
        $error = __('A username is required for registration.', 'frontendprofile');
    elseif ( username_exists($userdata['user_login']) )
        $error = __('Sorry, that username already exists!', 'frontendprofile');

    elseif ( !is_email($userdata['user_email']) )
        $error = __('You must enter a valid email address.', 'frontendprofile');
    elseif ( email_exists($userdata['user_email']) )
        $error = __('Sorry, that email address is already used!', 'frontendprofile');

else{
        $new_user = wp_insert_user( $userdata );

        wp_new_user_notification($new_user, $user_pass);

    }

}

// calling the header.php
get_header();?>

<div id="container">
    <div id="content">
            <div class="entry-content">
            </div>
        <!-- REGISTER FORM STARTS HERE -->

    <?php if ( is_user_logged_in() && !current_user_can( 'create_users' ) ) : ?>

        <p class="log-in-out alert">
        <?php printf( __('You are logged in as <a href="https://wordpress.stackexchange.com/questions/34060/%1$s" title="%2$s">%2$s</a>.  You don\'t need another account.', 'frontendprofile'), get_author_posts_url( $curauth->ID ), $user_identity ); ?> <a href="<?php echo wp_logout_url( get_permalink() ); ?>" title="<?php _e('Log out of this account', 'frontendprofile'); ?>"><?php _e('Logout &raquo;', 'frontendprofile'); ?></a>
        </p><!-- .log-in-out .alert -->

    <?php elseif ( @$new_user ) : ?>

        <p class="alert">
        <?php
            if ( current_user_can( 'create_users' ) )
                printf( __('A user account for %1$s has been created.', 'frontendprofile'), $_POST['user_name'] );
            else 
                printf( __('Thank you for registering, %1$s.', 'frontendprofile'), $_POST['user-name'] );
                printf( __('<br/>Please check your email address. That\'s where you\'ll recieve your login password.<br/> (It might go into your spam folder)', 'frontendprofile') );
        ?>
        </p><!-- .alert -->

    <?php else : ?>

        <?php if ( $error ) : ?>
            <p class="error">
                <?php echo $error; ?>
            </p><!-- .error -->
        <?php endif; ?>

        <?php if ( current_user_can( 'create_users' ) && $registration ) : ?>
            <p class="alert">
                <?php _e('Users can register themselves or you can manually create users here.', 'frontendprofile'); ?>
            </p><!-- .alert -->
        <?php elseif ( current_user_can( 'create_users' ) ) : ?>
            <p class="alert">
                <?php _e('Users cannot currently register themselves, but you can manually create users here.', 'frontendprofile'); ?>
            </p><!-- .alert -->
        <?php endif; ?>

        <?php if ( $registration || current_user_can( 'create_users' ) ) : ?>

        <form method="post" id="adduser" class="user-forms" action="http://<?php echo $_SERVER['HTTP_HOST'] . $_SERVER['REQUEST_URI']; ?>">


            <strong>Name</strong>

            <p class="form-username">
                <label for="user_name"><?php _e('Username (required)', 'frontendprofile'); ?></label>
                <input class="text-input" name="user_name" type="text" id="user_name" value="<?php if ( $error ) echo wp_specialchars( $_POST['user_name'], 1 ); ?>" />
            </p><!-- .form-username -->

            <p class="first_name">
                <label for="first_name"><?php _e('First Name', 'frontendprofile'); ?></label>
                <input class="text-input" name="first_name" type="text" id="first_name" value="<?php if ( $error ) echo wp_specialchars( $_POST['first_name'], 1 ); ?>" />
            </p><!-- .first_name -->

            <p class="last_name">
                <label for="last_name"><?php _e('Last Name', 'frontendprofile'); ?></label>
                <input class="text-input" name="last_name" type="text" id="last_name" value="<?php if ( $error ) echo wp_specialchars( $_POST['last_name'], 1 ); ?>" />
            </p><!-- .last_name -->

            <strong>Contact Info</strong>

            <p class="form-email">
                <label for="email"><?php _e('E-mail (required)', 'frontendprofile'); ?></label>
                <input class="text-input" name="email" type="text" id="email" value="<?php if ( $error ) echo wp_specialchars( $_POST['email'], 1 ); ?>" />
            </p><!-- .form-email -->

            <p class="form-submit">
                <input name="adduser" type="submit" id="addusersub" class="submit button" value="<?php if ( current_user_can( 'create_users' ) ) _e('Add User', 'frontendprofile'); else _e('Register', 'frontendprofile'); ?>" />
                <?php wp_nonce_field( 'add-user' ) ?>
                <input name="action" type="hidden" id="action" value="adduser" />
            </p><!-- .form-submit -->

        </form><!-- #adduser -->

        <?php endif; ?>

    <?php endif; ?>