Custom query form submission pagination

There is a question very similar to mine, which can be found here. The difference with my question is: my search form is on several pages (page.php and single.php) and I don’t use wp-pagenavi. I’ve tried the suggested solutions posted there but they didn’t work.

What is working

• The query displays the right results
• The query displays the right results after form submission
• Pagination shows with the correct links (ie example.com/overview/?paged=2 and example.com/city/boston/?paged=2)
• Pagination with the same settings works on search.php

What is not working: if I submit the form and click on any page but page 1 the URL does become example.com/overview/?paged=2 or example.com/city/boston/?paged=2 but no results are displayed.

Updated after the accepted answer:
Please read the notes in the code to see how the problem was solved.

— Below here the setup of my code

On page.php

if(is_page('overview')) {
    get_template_part('templates/overview', 'main');
}

On single.php (include the same file)

if(is_singular('cities')) {
    get_template_part('templates/overview', 'main');
}

On overview-main.php

include_once 'overview-search-input.php'; 
include_once 'overview-sidebar.php';

On overview-search-input.php (changed after UPDATE 3, original code below here)

   if(!empty($_GET['period'])) {
        $period = $_GET['period'];
   }
   else {
        $period = 'upcoming'; // Set to upcoming if nothing is selected
   }

   $query = search_v1($period, $paged); // Passing the $paged parameter in the query

   $paged = (get_query_var('paged')) ? get_query_var('paged') : 1; // The $paged parameter

On overview-sidebar.php

<form action="<?php echo get_permalink(); ?>" method="get"> 
<select name="period" placeholder="Select">
  <option value=""></option>
  <option value="all"></option>
  <option value="upcoming"></option>
</select>
</form>

in the function search_v1:

    $args = array(
        'post_type'         => 'events',
        'post_status'       => 'publish',
        'posts_per_page'    => 20,
        'paged'             => $paged, // This was passed into the function
        'meta_query'        => array($meta_query), // This is generated based on the $period input and it's working (tested)
        'meta_key'          => 'date_from',
        'orderby'           => 'meta_value',
        'order'             => 'ASC',
    );

    return new WP_Query($args);

Then on overview-main.php

   if($query->have_posts()): while ($query->have_posts()): $query->the_post();
      // Do stuff
    endwhile;

    create_pagination($query);

    wp_reset_postdata();
    endif;

And in the function create_pagination:

$query->query_vars['paged'] > 1 ? $current = $query->query_vars['paged'] : $current = 1;

$args = array(
    'base' => @add_query_arg('paged','%#%'),
    'format' => '?paged=%#%',
    'total' => $query->max_num_pages,
    'current' => $current,
    'show_all' => false,
    'end_size' => 1,
    'mid_size' => 2,
    'prev_next' => true,
    'prev_text' => '<i class="icon-chevron-left"></i>',
    'next_text' => '<i class="icon-chevron-right"></i>',
    'type' => 'list'
);

return paginate_links($args);

The problem was: Since both pagination functions work and display the correct pagination links I assume that’s not the problem. I think it lies in the custom query (since this pagination works on search.php when the results are generated once and not altered by form input).

The problem was solved: by using the ‘paged’ parameter and passing it into the query