WP Query paging not generating - php

Question

I have a seemingly simple WP query looking for posts on my site, and want these posts to be 'paged' accordingly. Below is my code, trying to achieve this, however all I am currently seeing is an empty .pager nav.
What is incorrect here? Help much appreciated.
Here is the live example.
PHP
<?php get_header(); ?>
<?php get_sidebar(); ?>
<main>
<?php
if ( get_query_var('paged') ) {
$paged = get_query_var('paged');
} elseif ( get_query_var('page') ) {
$paged = get_query_var('page');
} else {
$paged = 1;
}
$custom_query_args = array(
'post_type' => 'journal',
'posts_per_page' => 2,
'paged' => $paged,
'post_status' => 'publish',
'order' => 'DESC',
'orderby' => 'date'
);
$custom_query = new WP_Query( $custom_query_args );
if ( $custom_query->have_posts() ) :
while( $custom_query->have_posts() ) : $custom_query->the_post(); ?>
<article>
/* Article content here */
</article>
<?php endwhile; ?>
<?php if ($custom_query->max_num_pages > 1) : // custom pagination ?>
<nav class="pager">
<?php
$args = array(
'prev_next' => true,
'show_all' => true,
'type' => 'list'
);
echo paginate_links( $args );
?>
<ul class="page-prev-next">
<?php if( get_previous_posts_link() ) :
echo '<li class="prev-page">' . get_previous_posts_link( 'Prev' ) . '</li>';
endif; ?>
<?php if( get_next_posts_link() ) :
echo '<li class="next-page">' . get_next_posts_link( 'Next' ) . '</li>';
endif; ?>
</ul>
</nav>
<?php endif; ?>
<?php wp_reset_postdata(); ?>
<?php endif; ?>
</main>
<?php get_footer(); ?>