I am assuming you want these taxonomies data in each item of post_type JSON, but the logic is the same for taxonomies endpoint and JSON response.

I see two ways of achieving it.

1. filtering the data using rest_prepare_post_type filter (Reference) or
2. creating new field(s) using the register_rest_field function. (Reference)

I would recommend the second approach. This is the one I’ve used recently with a custom post type

Case 1: rest_prepare_post_type

In the first case, you would use the rest_prepare_post_type filter to get the data, generate the data you want, add it to the data array and return the filtered data. In this case, if the post type is “post”, the hook is rest_prepare_post.

/**
 * @summary    Filters post rest response
 *
 * @description    Filters post rest response removing or adding certain information
 * @param     object    $data       The response object.
 * @param     object    $post       The original post type object.
 * @param     object    $request    Request used to generate the response.
 * @return    object    $data       The filtered response object.
 */
function wpse_283452_rest_prepare_post($data, $post, $request) {
    $_data = $data->data;
    // say you want to remove author from the response
    unset($_data['author']);
    // if you want to ADD fields, you would do the logic here and...
    // add it to the $_data array
    $data->data = $_data;
    // and finally return it
    return $data;
}
add_filter('rest_prepare_post', 'wpse_283452_rest_prepare_post', 10, 3);

If you want to the same for taxonomies, you would use rest_prepare_taxonomy/ (reference)

Case 2: register_rest_field (recommended)

You will register a new field with register_rest_field function and populate it the way you want using a callback

/**
 * Registers a new field on an existing WordPress object type.
 *
 * @since 4.7.0
 *
 * @global array $wp_rest_additional_fields Holds registered fields, organized
 *                                          by object type.
 *
 * @param string|array $object_type Object(s) the field is being registered
 *                                  to, "post"|"term"|"comment" etc.
 * @param string $attribute         The attribute name.
 * @param array  $args {
 *     Optional. An array of arguments used to handle the registered field.
 *
 *     @type string|array|null $get_callback    Optional. The callback function used to retrieve the field
 *                                              value. Default is 'null', the field will not be returned in
 *                                              the response.
 *     @type string|array|null $update_callback Optional. The callback function used to set and update the
 *                                              field value. Default is 'null', the value cannot be set or
 *                                              updated.
 *     @type string|array|null $schema          Optional. The callback function used to create the schema for
 *                                              this field. Default is 'null', no schema entry will be returned.
 * }
 */
register_rest_field(
    'post',
    'wpse_283452_taxonomies',
    array(
        'get_callback' => function ($object, $field_name, $request) {
            return wpse_283452_taxonomies_callback($object);
        },
    )
);

/**
 * @summary    Return and array with taxonomy info
 *
 * @description    Returns an array with taxonomy information in a custom way
 * @param     object    $object            The rest item original object
 * @return    array     $custom_taxonom    an array with taxonomy terms with their information in a custom way
 */
function wpse_283452_taxonomies_callback($object) {
    $post_id = $object['id'];
    $terms = wp_get_post_terms(
        $post_id,
        'business_type',
        array(
            'fields' => 'all',
        )
    );
    // here you should construct an array with the information in $terms in the way/structure you want
    return $terms;
}

For taxonomies, you would use the taxonomy slug instead of ‘post’ as the 1st argument on register_rest_field.

That’s it.

Let me know if works.