function get_pagination_in_json( $post_response, $post, $context ) {

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

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

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

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

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

    return $post_response;
}
add_filter( 'json_prepare_post', 'get_pagination_in_json', 10, 3 );