According to your question, you need to take two main steps:

Step 1: Instead of 1, 2, 3, 4...Last Number you need 1, 2, 3...19, 20, Last Number

Step 2: You need to add 0 prefix in the pagination numbering when the number is less than 10, that is: 01, 02, 03...19, 20, Last Number

This is definitely possible using paginate_links() function.

Solution for Step 1:

To make sure 3 pagination numbers show up from the two ends, you need to use the attribute: 'end_size' => 3

Solution for Step 2:

To make sure numbers less than 10 are prefixed with 0, you need to use the filter hook: number_format_i18n

Final CODE:

First, place the following function somewhere in your functions.php file:

    /* Filter function to be used with number_format_i18n filter hook */
    if( ! function_exists( 'wpse255124_zero_prefix' ) ) :
    function wpse255124_zero_prefix( $format ) {
        $number = intval( $format );
        if( intval( $number / 10 ) > 0 ) {
            return $format;
        }
        return '0' . $format;
    }
    endif;

Then, use the following CODE instead of the one you’ve provided in your question:

    $format = $GLOBALS['wp_rewrite']->using_index_permalinks() && ! strpos( $pagenum_link,
            'index.php' ) ? 'index.php/' : '';
    $format .= $GLOBALS['wp_rewrite']->using_permalinks() ? user_trailingslashit( 'page/%#%',
        'paged' ) : '?paged=%#%';

    add_filter( 'number_format_i18n', 'wpse255124_zero_prefix' );

    /*Set up paginated links.*/
    $links = paginate_links( array(
        'base'      => $pagenum_link,
        'format'    => $format,
        'total'     => $wp_query->max_num_pages,
        'current'   => $paged,
        'mid_size'  => 1,
        'end_size'  => 3,
        'after_page_number' => '.',
        'add_args'  => array_map( 'urlencode', $query_args ),
        'prev_text' => '',
        'next_text' => '',
        'type'      => 'list'
    ) );

    remove_filter( 'number_format_i18n', 'wpse255124_zero_prefix' );

Now you should get your expected pagination.