As I'm also looking for a solution to this issue I've come a bit further and maybe it will get you in the right direction.

To load a menu in Drupal 8 you can put this in a `theme_preprocess_page` in your `.theme`-file.

      $menu_tree = \Drupal::menuTree();
      $menu_name = 'main';
    
      // Build the typical default set of menu tree parameters.
      $parameters = $menu_tree->getCurrentRouteMenuTreeParameters($menu_name);
    
      // Load the tree based on this set of parameters.
      $tree = $menu_tree->load($menu_name, $parameters);
    
      // Transform the tree using the manipulators you want.
      $manipulators = array(
        // Only show links that are accessible for the current user.
        array('callable' => 'menu.default_tree_manipulators:checkAccess'),
        // Use the default sorting of menu links.
        array('callable' => 'menu.default_tree_manipulators:generateIndexAndSort'),
      );
      $tree = $menu_tree->transform($tree, $manipulators);
    
      // Finally, build a renderable array from the transformed tree.
      $menu = $menu_tree->build($tree);
      $menu_html = \Drupal::service('renderer')->renderPlain($menu);

I have not yet worked out how to merge different menus into one big menu but I will let you know if I come up with a solution.
You can find out more about the code above at [Drupal 8.2.x » menu.api.php][1] which is where I found it.


Ok, I now have found the solution which I found in the module [Responsive and off-canvas menu][2]

      $menus = explode(',', "main-menu,footer-menu,utility-menu");
    
      $combined_tree = array();
      $menu_tree = \Drupal::menuTree();
      $parameters = $menu_tree->getCurrentRouteMenuTreeParameters(trim($menus[0]));
      $manipulators = array(
        // Show links to nodes that are accessible for the current user.
        array('callable' => 'menu.default_tree_manipulators:checkNodeAccess'),
        // Only show links that are accessible for the current user.
        array('callable' => 'menu.default_tree_manipulators:checkAccess'),
        // Use the default sorting of menu links.
        array('callable' => 'menu.default_tree_manipulators:generateIndexAndSort'),
      );
      // Force the entire tree to be build by setting expandParents to an
      // empty array.
      $parameters->expandedParents = array();
      // Iterate over the menus and merge them together.
      foreach($menus as $menu_name) {
        $tree_items = $menu_tree->load(trim($menu_name), $parameters);
        $tree_manipulated = $menu_tree->transform($tree_items, $manipulators);
        $combined_tree = array_merge($combined_tree, $tree_manipulated);
      }
    
      $menu = $menu_tree->build($combined_tree);
    
      $vars['mobile_menu'] = array(
        '#prefix' => '<div class="off-canvas-wrapper"><div id="off-canvas">',
        '#suffix' => '</div></div>',
        '#markup' => \Drupal::service("renderer")->renderRoot($menu),
      );


If you put this in your `theme_preprocess_page` in your `.theme`-file you could then in your `page.html.twig` print the twig variable `{{ mobile_menu }}`.

  [1]: https://api.drupal.org/api/drupal/core!lib!Drupal!Core!Menu!menu.api.php/group/menu/8.2.x#sec_rendering
  [2]: https://www.drupal.org/project/responsive_menu