Basic plugin which adds a nav menu to Reader mode templates in the AMP plugin. When in Reader mode, upon activation a new “AMP Reader Sidebar” nav menu location is available for assigning the desired nav menu. Temporary until closed: https://github.com/ampproject/amp-wp/issues/2044

amp-reader-nav-menu.php

<?php
/**
 * Plugin Name: AMP Reader Nav Menu
 *
 * @package   AMP_Reader_Nav_Menu
 * @author    Weston Ruter, Google
 * @license   GPL-2.0-or-later
 * @copyright 2019 Google Inc.
 *
 * @wordpress-plugin
 * Plugin Name: AMP Reader Nav Menu
 * Description: Add nav menu for Reader mode templates. If in Reader mode, upon activation you'll see a new nav menu location to which you can <a href="nav-menus.php">assign the desired nav menu</a>.
 * Plugin URI: https://gist.github.com/westonruter/b22cfae13f0862d5b9716c776c60f882
 * Version: 0.1.1
 * Author: Weston Ruter, Google
 * Author URI: https://weston.ruter.net/
 * License: GNU General Public License v2 (or later)
 * License URI: http://www.gnu.org/licenses/gpl-2.0.html
 * Gist Plugin URI: https://gist.github.com/westonruter/b22cfae13f0862d5b9716c776c60f882
 */

namespace AMP_Reader_Nav_Menu;

const NAV_MENU_LOCATION = 'amp_reader_sidebar';

// Register the nav menu location if not in Standard/Transitional modes.
add_action(
	'init',
	function() {
		if ( ! current_theme_supports( 'amp' ) ) {
			register_nav_menu( NAV_MENU_LOCATION, __( 'AMP Reader Sidebar', 'amp-reader-nav-menu' ) );
		}
	}
);

add_action(
	'amp_post_template_css',
	function () {
		printf(
			'
			#toggleNavMenu { font-size: 22px; color: white; position: absolute; top: .5em; %1$s: -15px; padding: 0 30px; background: none; border: none; }
			.amp-site-title { position:relative; %1$s: 30px; }
			#navMenu { padding: 1em; } 
			#navMenu li { display: block; }
			',
			is_rtl() ? 'right' : 'left'
		);
	}
);

add_filter(
	'amp_post_template_file',
	function ( $template_file ) {
		if ( basename( $template_file ) === 'header-bar.php' ) {
			$template_file = __DIR__ . '/header-bar.php';
		}
		return $template_file;
	}
);

add_filter(
	'amp_post_template_data',
	function( $data ) {
		$data['amp_component_scripts'] = array_merge(
			$data['amp_component_scripts'],
			array(
				'amp-sidebar' => true,
			)
		);
		return $data;
	}
);

header-bar.php

<?php
/**
 * Header bar template part.
 *
 * This is a forked version of the header-bar.php in the AMP plugin.
 *
 * @link https://github.com/ampproject/amp-wp/blob/e63d2c3034d079f1a1b5bb2f0a2d1e0a221cadb9/templates/header-bar.php
 * @package AMP_Reader_Nav_Menu
 */

namespace AMP_Reader_Nav_Menu;

/**
 * Context.
 *
 * @var \AMP_Post_Template $this
 */

?>
<header id="top" class="amp-wp-header">
	<div>
		<?php if ( has_nav_menu( NAV_MENU_LOCATION ) ) : ?>
			<button id="toggleNavMenu" type="button" on="tap:navMenu.open" aria-label="<?php esc_html__( 'Open nav menu', 'amp-reader-nav-menu' ); ?>">&#9776;</button>
		<?php endif; ?>

		<a href="<?php echo esc_url( $this->get( 'home_url' ) ); ?>">
			<?php $site_icon_url = $this->get( 'site_icon_url' ); ?>
			<?php if ( $site_icon_url ) : ?>
				<amp-img src="<?php echo esc_url( $site_icon_url ); ?>" width="32" height="32" class="amp-wp-site-icon"></amp-img>
			<?php endif; ?>
			<span class="amp-site-title">
				<?php echo esc_html( wptexturize( $this->get( 'blog_name' ) ) ); ?>
			</span>
		</a>

		<?php $canonical_link_url = $this->get( 'post_canonical_link_url' ); ?>
		<?php if ( $canonical_link_url ) : ?>
			<?php $canonical_link_text = $this->get( 'post_canonical_link_text' ); ?>
			<a class="amp-wp-canonical-link" href="<?php echo esc_url( $canonical_link_url ); ?>">
				<?php echo esc_html( $canonical_link_text ); ?>
			</a>
		<?php endif; ?>
	</div>
</header>

<?php if ( has_nav_menu( NAV_MENU_LOCATION ) ) : ?>
	<?php
	printf(
		'<amp-sidebar id="navMenu" layout="nodisplay" side="%s">%s</amp-sidebar>',
		esc_attr( is_rtl() ? 'right' : 'left' ),
		wp_nav_menu(
			[
				'theme_location' => NAV_MENU_LOCATION,
				'echo'           => false,
				'depth'          => 1,
			]
		)
	);
	?>
<?php endif; ?>

i think you have to change
'location' => NAV_MENU_LOCATION,
with
'theme_location' => NAV_MENU_LOCATION,

@peppolone Yes, I think you are right. Updated. Thanks.

Installation instructions: https://gist.github.com/westonruter/6110fbc4bef0c4b8c021a112012f7e9c

Hey @westonruter thank you for this solution, it worked perfectly in my project.