Display posts grouped by post type in taxonomy.php

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Short version

Within your loop call: get_template_part( 'loop', get_post_type() );

You need now a loop.php as general fallback template (the default view).
You can generate as many post_type specific templates as you need, e.g. loop-newpost.php and loop-book.php.

But you are overthinking it a bit and adding another query on top the main query (page load?). You can add full template support to your theme with the following.

A better apporach

Add your CPTs.

function add_my_custom_post_type() {
    $args = array(
        'menu_position' => 8,
        'menu_icon' => 'dashicons-admin-post',
        'public' => true,
        'publicly_queryable' => true,
        'query_var' => false,
        'capability_type' => 'post',
        'has_archive' => true,
        'rewrite' => array(
            'slug' => 'new-post',
            'feeds' => true,
            'pages' => true,
            'with_front' => true,
        ),
        'taxonomies' => array(
            'author'
        ),
        'supports' => array(
            'title',
            'editor',
            'thumbnail',
            'excerpt'
        )
    );
    register_post_type( 'newpost', $args );
    $args['labels']['name'] = __('Books', 'my_textdomain');
    $args['labels']['all_items'] = __('All Books', 'my_textdomain');
    $args['labels']['singular_name'] = __('Book', 'my_textdomain');
    $args['rewrite']['slug'] = 'book';
    register_post_type( 'book', $args );
}
add_action('init', 'add_my_custom_post_type');

Add your custom taxonomy.

Attention: author is a slug used by WordPress, so you need to be creative: tax_author for the taxonomy name & query_var and the-author for the slug.

function add_my_custom_taxonomy() {
    register_taxonomy(
        'tax_author',
        array(
            'newpost',
            'book'
        ),
        array(
            'hierarchical' => false,
            'labels' => array(
                'name' => __( 'Authors', 'my_textdomain' ),
                'singular_name' => __( 'Author', 'my_textdomain' ),
                'menu_name' => __( 'Author', 'my_textdomain' ),
            ),
            'show_ui' => true,
            'show_admin_column' => true,
            'update_count_callback' => '_update_post_term_count',
            'query_var' => true,
            'rewrite' => array(
                'slug' => 'the-author',
                'with_front' => true,
                'hierarchical' => false
            )
        )
    );
}
add_action('init', 'add_my_custom_taxonomy');

Add the CPT to the main queries in questions with the pre_get_posts hook

function add_my_custom_post_type_to_query( $query ) {
    if ( $query->is_main_query() && $query->is_home() && !is_admin() /* No filtering of CPTs in the backend */) {
        $query->set( 'post_type', array('newpost', 'book') );
    }
}

add_action( 'pre_get_posts', 'add_my_custom_post_type_to_query' );

Edit: You don’t need this for taxonomies, but if you want to add your CPT to the main query, use this hook. — Thanks to @Pieter Goosen for the clarification.

Within your taxonomy-tax_author.php template file you can now use the main loop:

if (have_posts()) while (have_posts()):
    the_post();
    get_template_part( 'loop', get_post_type() );

You need now a loop.php as general fallback template (the default view).
You can generate as many post_type specific templates as you need, e.g. loop-newpost.php and loop-book.php.

Update:
If you want to order the query by post_type you need to hook into the pre_get_posts again

function order_my_custom_taxonomy( $query ) {
    if ( $query->is_main_query() && $query->is_tax('tax_author') && !is_admin() /* No filtering of CPTs in the backend */) {
        $query->set( 'orderby', 'type' );
        $query->set( 'order', 'ASC' );
    }
}

add_action( 'pre_get_posts', 'add_my_custom_post_type_to_query' );

For detailed information about how to filter and order follow the provided links.

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