next_posts_link does not work when querying a specific category

You’re mixing query string and array parameters.

They should either be a single string, like:

$wp_query = new WP_Query( 'cat=13&paged=' . $paged );

or an array of key/value parameters, like:

$wp_query = new WP_Query(
    array(
        'cat' => 13,
        'paged' => $paged
    )
);