DocSheldon on "Edit buttons on widgets"

I recently updated one of my UK sites to version 4.5.2–en_GB, whereupon I noticed that the traditional drag & drop method of setting up widgets in the backend had disappeared, leaving me with this:
http://www.awesomescreenshot.com/0c75vzc02e
A colleague corroborated that recent changes had affected widgets this way, but that her version didn't have the edit button. Another colleague provided me with this screenshot:
http://screencast.com/t/RVjSspmz658
Both these colleagues are US-based and wouldn't be running the en_GB version, so some differences might make sense to me. However, on several of my US-based sites, I'm running the US version of 4.5.2 and the traditional drag & drop widget placement still exists.
Can anyone shed any light on this? I've investigated possible WP install issues (replaced the wp-admin and wp-includes folders), removed all plugins... I even tried changing themes to 2015 to see if somehow some changes I'd implemented in my child theme were at fault.
Am I chasing my tail? Is this just something that's unique to the en_GB version?

I just updated another of my UK sites to 4.5.2 en_GB version and the widgets pages there still shows the traditional drag & drop style. So I guess that shoots down that explanation.

make sure to have javascript enabled in your browser.

It's enabled, Michael. Thanks.

It's enabled, Michael. Thanks.

No, Michael... that was one of the first things I checked, last night... it was enabled all along.

I don't know that this offers anything, Michael, but this is the version.php file from the site:

<?php
/** * The WordPress version string * * @global string $wp_version */
$wp_version = '4.5.2';
/** * Holds the WordPress DB revision, increments when changes are made to the WordPress DB schema. * * @global int $wp_db_version */
$wp_db_version = 36686;
/** * Holds the TinyMCE version * * @global string $tinymce_version */
$tinymce_version = '4310-20160418';
/** * Holds the required PHP version * * @global string $required_php_version */
$required_php_version = '5.2.4';
/** * Holds the required MySQL version * * @global string $required_mysql_version */
$required_mysql_version = '5.0';