Get terms ordered by post date

Basically, what you are doing right now is this:

1. Get all the terms from industry taxonomy
2. For each term do a WP_Query to get the newest posts, iterate through them to sort the terms, creating a new array with term ids
3. Sort and slice the new array
4. For each element in new array of terms ids do again a WP_Query to get the newest posts, and iterate through them

That’s a lot of WP_Queryies. Let’s try to lower that number.

We can do it like this:

1. Get all the terms from industry taxonomy
2. For each term do a WP_Query to get the newest posts, iterate through them to sort the terms, creating a new array
3. New array will be keyed with a date in U format, and contain both the term object and the array with post objects, so we won’t have to query them again.

Here is my suggested code:

$taxonomy = 'industry'; //taxonomy name
$terms_to_display = 5; //number of terms to display
$posts_per_term = 4; // number of posts per term to display
$terms = get_terms( array('taxonomy' => $taxonomy, 'hide_empty' => true ) );
$news_items = [];

foreach ( $terms as $term ) {

    $args = array (
        'post_type'         => 'news',
        'posts_per_page'    => $posts_per_term,
        'orderby'           => 'date',
        'order'             => 'DESC',
        'tax_query' => array (
            array (
                'taxonomy' => $taxonomy,
                'field'    => 'id',
                'terms'    => array( $term->term_id ),
            ),
        ),
    );

    $news_query = new WP_Query( $args );

    // ignore terms with not enough posts
    if ( $news_query->post_count < $posts_per_term ) {
        continue;
    }

    $news_posts = $news_query->get_posts();

    // get date of newest post in this term
    $newest_post_date = get_the_date( "U", $news_posts[0]->ID );

    $news_items[$newest_post_date] = array(
        'term' => $term,
        'news' => $news_posts,
    );
}

krsort( $news_items ); // sort descending by keys

$news_items = array_slice( $news_items, 0, $terms_to_display );

wp_reset_query();

I have made a few more changes as well:
1. $terms is now array of the WP_Term objects, not just ids
2. Made the $news_query->post_count < $posts_per_term check slightly more readable

And how to use:

foreach ( $news_items as $item ) {

    echo "<h2>" . $item['term']->name . "</h2>";

    echo "<ul>";

    foreach ( $item['news'] as $news_post ) {
        setup_postdata( $news_post );

        echo "<li>" . get_the_title() . "</li>";
    }

    echo "</ul>";
}

wp_reset_query();

This way, we have lowered the number of WP_Queryies to the number of non-empty terms in the industry taxonomy, and made the resulting array much more useful.