Wordpress: pagination with custom posts. The key is to use the DEFAULT loop, with the necessary data for the loop coming from the name of the file itself, in this case: taxonomy-skill_tag.php. The WP_Query stuff will work only if default post number display options is set to 1

functions.php

// fix bug when custom posts archive pagination does not work on page 2 and subsequent. 
// This is the hack which works when WP_Query (not default loop) is used
$option_posts_per_page = get_option( 'posts_per_page' );
add_action( 'init', 'my_modify_posts_per_page', 0);
function my_modify_posts_per_page() {
	add_filter( 'option_posts_per_page', 'my_option_posts_per_page' );
}
function my_option_posts_per_page( $value ) {
	global $option_posts_per_page;
	if ( is_tax( 'skill_tag') ) {
		return 1;
	} else {
		return $option_posts_per_page;
	}
}

taxonomy-skill_tag.php

<?php get_header(); ?>


  <?php $term = get_term_by( 'slug', get_query_var( 'term' ), get_query_var( 'taxonomy' ) );
  ?>

  <div id="container">
    <div id="content" role="main">
      <header class="page-heading section">
        <h1 class="page-title"><?php echo $term->name; ?></h1>
      </header><!-- .entry-header -->


        <div class="entries-wrapper">
        <?php while ( have_posts() ) : the_post(); ?>

          <div class="post type-post entry">
            <img class="entry__image" src="<?php the_field('skill_image'); ?>"/>
            <div class="entry__details-wrapper">
              <h3 class="entry__title">
                <?php
                if ($term->term_id != 250 && $term->term_id != 252) : ?>
                <a href="<?php echo get_permalink(); ?>"
                title="<?php the_title(); ?>" rel="bookmark">
                  <?php the_title(); ?>
                </a>
              <?php else : the_title(); ?>

                <?php endif; ?>
              </h3>
              <div class="entry__summary">
                <?php if ( get_field('skill_completion_status') ) :?>
                  <p class="label label-future-skill"><?php the_field('skill_completion_status') ?></p>
                <?php endif; ?>
                <?php the_field('short_description'); ?>
              </div><!-- .entry-summary -->
            </div>
          </div>
        <?php endwhile;  ?>

        </div>
      <?php
       wp_reset_query();
       ?>
       <div class="pagination">
        <?php
        echo paginate_links( array(
            'base'         => str_replace( 999999999, '%#%', esc_url( get_pagenum_link( 999999999 ) ) ),

            'current'      => max( 1, get_query_var( 'paged' ) ),
            'format'       => '?paged=%#%',
            'show_all'     => false,
            'type'         => 'plain',
            'end_size'     => 2,
            'mid_size'     => 1,
            'prev_next'    => true,
            'prev_text'    => sprintf( '<i></i> %1$s', __( 'Previous', 'text-domain' ) ),
            'next_text'    => sprintf( '%1$s <i></i>', __( 'Next', 'text-domain' ) ),
            'add_args'     => false,
            'add_fragment' => '',
        ) );
        ?>
        </div>

    </div><!-- #content -->
  </div><!-- #container -->

<?php get_footer(); ?>

taxonomy-skill_tag.php (with custom loop instead of default one)

<?php get_header(); ?>


  <?php $term = get_term_by( 'slug', get_query_var( 'term' ), get_query_var( 'taxonomy' ) );
  $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;

  $args = array(
    'post_type' => 'skills',
    // 'nopaging' => true,
    'orderby' => 'menu_order',
    'order' => 'ASC',
    'posts_per_page' => 2,
    'posts_per_archive_page' => 2,
    'paged' => $paged,
    'tax_query' => array(
      array(
        'taxonomy' => 'skill_tag',
        'field' => 'name',
        'terms' => $term->name
      )
    )
  );
  if ($term->term_id == 250 || $term->term_id == 252) {
    $args['meta_key']	= 'skill_completion_status';
    $args['orderby']  = 'meta_value';
    $args['order']    = 'DESC';
  }
  $currentlyViewedSkillGroup = new WP_Query( $args );
  ?>

  <div id="container">
    <div id="content" role="main">
      <header class="page-heading section">
        <h1 class="page-title"><?php echo $term->name; ?></h1>
      </header><!-- .entry-header -->

      <?php if ($currentlyViewedSkillGroup->have_posts()) : ?>
        <div class="entries-wrapper">
        <?php while ( $currentlyViewedSkillGroup->have_posts() ) : $currentlyViewedSkillGroup->the_post(); ?>

          <div class="post type-post entry">
            <img class="entry__image" src="<?php the_field('skill_image'); ?>"/>
            <div class="entry__details-wrapper">
              <h3 class="entry__title">
                <?php
                if ($term->term_id != 250 && $term->term_id != 252) : ?>
                <a href="<?php echo get_permalink(); ?>"
                title="<?php the_title(); ?>" rel="bookmark">
                  <?php the_title(); ?>
                </a>
              <?php else : the_title(); ?>

                <?php endif; ?>
              </h3>
              <div class="entry__summary">
                <?php if ( get_field('skill_completion_status') ) :?>
                  <p class="label label-future-skill"><?php the_field('skill_completion_status') ?></p>
                <?php endif; ?>
                <?php the_field('short_description'); ?>
              </div><!-- .entry-summary -->
            </div>
          </div>
        <?php endwhile; ?>
          <div class="pagination">
          <?php
          echo paginate_links( array(
              'base'         => str_replace( 999999999, '%#%', esc_url( get_pagenum_link( 999999999 ) ) ),
              'total'        => $currentlyViewedSkillGroup->max_num_pages,
              'current'      => max( 1, get_query_var( 'paged' ) ),
              'format'       => '?paged=%#%',
              'show_all'     => false,
              'type'         => 'plain',
              'end_size'     => 2,
              'mid_size'     => 1,
              'prev_next'    => true,
              'prev_text'    => sprintf( '<i></i> %1$s', __( 'Newer Posts', 'text-domain' ) ),
              'next_text'    => sprintf( '%1$s <i></i>', __( 'Older Posts', 'text-domain' ) ),
              'add_args'     => false,
              'add_fragment' => '',
            ) );
          ?>
          </div>
        </div>
      <?php endif;
      wp_reset_postdata(); ?>

    </div><!-- #content -->
  </div><!-- #container -->

<?php get_footer(); ?>