Taxonomy question

The short answer is to get your taxonomy terms using:

wp_get_post_terms( $post->ID, 'yourTaxonomyName' );

This will return an array of WP Term Objects. Here’s how you would get the first term name:

$terms = wp_get_post_terms( $post->ID, 'yourTaxonomyName' );
$first_term_name = $terms[0]->name;

You can learn more about this method in the WordPress Codex:
https://codex.wordpress.org/Function_Reference/wp_get_post_terms

Hope this helps 🙂

Updated:

private function get_product_name( $product ) {
    // Get all the country terms associated with $product.
    $terms = wp_get_post_terms($product->ID, 'country');

    // Set the empty $country string.
    $country = '';

    // Check if there are country terms available.
    if($terms){

        // If there are terms then use the first one to populate the $country variable. 
        $country = '</br><i>' . $terms[0]->name . '</i>';
    }
    // Add $country to the $name variable.
    $name = wcpt_get_name( $product ) . $country . '</td>';

    // This part could rewrite $name which would remove $country.
    if ( array_intersect( [ 'all', 'name' ], $this->args->links ) ) {
        $name = WCPT_Util::format_product_link( $product, $name );
    }

    return apply_filters( 'wc_product_table_data_name', $name, $product );
}