How to query 2 custom post types that need to share a slug?

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Note that a filter callback must always return something which is commonly the first parameter passed to the callback. So for example your overwriteQueryVars() function, it must always return the $query because request is a filter hook.

Secondly, instead of overwriting the entire query arguments (e.g. by doing $query = array( ... ) or $query->query_vars = $new_query;), you should only change the ones need to be changed in the existing query arguments, e.g. $query['post_type'] = $postType;.

But anyway, try this which worked well for me:

add_filter( 'request', 'overwriteQueryVars' );
function overwriteQueryVars( $query_vars ) { // I renamed $query to $query_vars.
    global $wp;

    /*
     * 1. Check if the current page URL path begins with library/authors/ as in example.com/library/authors/
     * 2. Then check if the post_type is exactly "authors".
     * 3. Finally, check if the "authors" is in the form of <slug>/<slug>.
     */
    if ( preg_match( '#^library/authors/#', $wp->request )        &&
        isset( $query_vars['post_type'], $query_vars['authors'] ) &&
        'authors' === $query_vars['post_type']                    &&
        preg_match( '#^([^/]+)/([^/]+)$#', $query_vars['authors'] )
    ) {
        $works_slug = explode( "https://wordpress.stackexchange.com/", $query_vars['authors'] )[1];

        // Find "works" post with the above slug.
        $ids = get_posts( array(
            'post_type' => 'works',
            'name'      => $works_slug,
            'fields'    => 'ids',
        ) );

        // If the post exists, then load it.
        if ( ! empty( $ids ) ) {
            $query_vars['post_type'] = 'works';
            $query_vars['name'] = $query_vars['works'] = get_page_uri( $ids[0] );

            unset( $query_vars['authors'] );
        }
    }

    return $query_vars; // ALWAYS RETURN IT
}

Additionally, in the post_type_link filter callback, I would:

  1. Change the 'library/authors/' . $author . "https://wordpress.stackexchange.com/" . $post->post_name to 'library/authors/' . $author . "https://wordpress.stackexchange.com/" . ( $leavename ? '%postname%' : $post->post_name ) to ensure the post permalink (the slug part) is editable via the post editing screen (in wp-admin).

  2. Change the $author = basename(get_permalink($author_id)); to $author = $author_id ? get_post_field( 'post_name', $author_id ) : '';.

Update

Sorry, I forgot about preview URLs (e.g. for posts with the status draft or pending) where the URL is not “pretty” and looks like https://example.com/?post_type=works&p=123&preview=true.

So to ensure the URL returned is correct, in your my_website_filter_post_type_link() function, add this at the very top:

if ( $sample || wp_force_plain_post_permalink( $post ) ) {
    return $post_link;
}

I.e. Add it above the line having the switch( $post->post_type ).

Also (to other readers), if your works post type is not hierarchical, replace the get_page_uri( $ids[0] ) with $works_slug.