Add pagination to for each taxonomy terms

That pagination is for posts, whereas you are trying to paginate terms.

One approach is to create the pagination for yourself:

$term_id = $term->term_id;
$taxonomy_name = $term->taxonomy;
$termchildren = get_term_children($term_id, $taxonomy_name);

$per_page = 10;
$page  = get_query_var('page');
$paged = get_query_var('paged');
$total_pages = ceil(count($termchildren) / $per_page);

$current_page = ($paged ? $paged : ($page ? $page : 1));
$offset = (($current_page - 1) * $per_page);

$term_children = array_slice($term_children, $offset, $per_page);

Untested, but that’s a starting point. You can use $total_pages and $current_page to build the pagination links.