add_action( 'rest_api_init', 'slug_register_pagination' );

function slug_register_pagination() {
    register_rest_field( 'post',
        'pagination',
        array(
            'get_callback'    => 'get_pagination_in_json',
            'update_callback' => null,
            'schema'          => null,
        )
    );
}

function get_pagination_in_json($post, $field_name, $request) {

    // Ensure global post is correctly set
    $old_post = $GLOBALS['post'];
    $GLOBALS['post'] = get_post($post['id']);
    $post = $GLOBALS['post'];
    setup_postdata($post);

    $previous_post = get_adjacent_post(true, '', true, 'category' );
    $next_post = get_adjacent_post(true, '', false, 'category' );

    if ( is_a( $previous_post, 'WP_Post' ) ) {
        $post_response['previous'] = $previous_post->post_name;
    }

    if ( is_a( $next_post, 'WP_Post' ) ) {
        $post_response['next'] = $next_post->post_name;
    }

    // Reset global post to its old value
    $GLOBALS['post'] = $old_post;

    return $post_response;
}