Solution 1:

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( '&laquo; PREV', $cpt_query->max_num_pages) ?></li> 
        <li><?php next_posts_link( 'NEXT &raquo;', $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('&laquo; Newer') ?>
    <?php next_posts_link('Older &raquo;') ?>
</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>

Solution 2:

I think you put <?php wp_reset_query(); ?> in the wrong place.. shouldn't it be next or after pagination codes?

something like this

<?php endwhile; ?>
<?php else: ?>
<?php wp_reset_query(); ?>