There are 3 ways that I would suggest for pagination with a custom post wp_query. Unfortunately to this day there isn't a lot of good information about this out there, or at least what is out there is unclear in some cases. Hopefully this helps!
Note, you also did have the wp_reset_postdata() in the wrong place, but even still more is needed to get it to work correctly.
Option 1 - use max_num_pages variable
<?php
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$args = array(
'posts_per_page' => 1,
'paged' => $paged,
'post_type' => 'cpt_type'
);
$cpt_query = new WP_Query($args);
?>
<?php if ($cpt_query->have_posts()) : while ($cpt_query->have_posts()) : $cpt_query->the_post(); ?>
//Loop Code Here...
<?php endwhile; endif; ?>
<nav>
<ul>
<li><?php previous_posts_link( '« PREV', $cpt_query->max_num_pages) ?></li>
<li><?php next_posts_link( 'NEXT »', $cpt_query->max_num_pages) ?></li>
</ul>
</nav>
You'll see above, a slightly different format for previous_posts_link
and next_posts_link
which now access the max_num_pages
variable. Be sure to use your own query variable name when accessing max_num_pages
. Notice I use $cpt_query since that is the variable for my query example.
Option 2 - temporarily use the $wp_query variable for your loop query
This is what a lot of folks recommend, but be careful to asign the $wp_query variable to a temp variable and re-assign it or you will run in to all kinds of troubles. Which is why I recommend Option #1. As noted on CSS Tricks, you can do something like this:
<?php
$temp = $wp_query;
$wp_query = null;
$wp_query = new WP_Query();
$wp_query->query('showposts=6&post_type=news'.'&paged='.$paged);
while ($wp_query->have_posts()) : $wp_query->the_post();
?>
<!-- LOOP: Usual Post Template Stuff Here-->
<?php endwhile; ?>
<nav>
<?php previous_posts_link('« Newer') ?>
<?php next_posts_link('Older »') ?>
</nav>
<?php
$wp_query = null;
$wp_query = $temp; // Reset
?>
Option 3 - use WP-pagenavi plugin
Just as another option what you can do instead is use the WP-pagenavi plugin, and setup your query as in Option #1. But make one change in the code, remove everything within the element and replace with this function, once you have installed the plugin. So you'll end with:
<nav>
<?php wp_pagenavi( array( 'query' => $cpt_query ) ); ?>
</nav>