User Archive page for Custom Post Type – Pagination Issue

have you already tried this way?

<section class="posts-container">
    <div class="container">
        <h2 style="text-transform: uppercase;">Posts from <?php echo get_the_author_meta( 'display_name', $author ); ?></h2>

        <?php
        $args = array(
            'post_type'      => array( 'style', 'travel' ),
            'posts_per_page' => -1,
            'author'         => $author
        );

        $author_videos = new WP_Query( $args );
        if ( $author_videos->have_posts() ) : while ( $author_videos->have_posts() ) : $author_videos->the_post(); ?>

            <?php
              // I think you should use archive rather than single
              // get_template_part('partials/archive', 'post');
                 get_template_part('partials/single', 'post');
            ?>

        <?php endwhile; ?>

            <div class="clear"></div>

            <div class="nav-previous alignleft">
                <?php next_posts_link( 'Older posts' ); ?>
            </div>
            <div class="nav-next alignright">
                <?php previous_posts_link( 'Newer posts' ); ?>
            </div>

        <?php wp_reset_postdata(); ?>

        <?php endif; ?>

    </div>
</section>