Taxonomy autocomplete returns "connection reset"
So after much frustration, I've finally got to the bottom of this. The answer is not within Drupal, but rather in my cPanel setup. I'll post the answer here because it may save someone else the headaches I've had.

The problem was that my cPanel hosting had xVarnish enabled. I'm not sure if it was suddenly added by a package upgrade, or if the behaviour changed, or if its effect only kicked in after a certain time span, but I certainly didn't switch it on! In any case, disabling xVarnish not only fixed this problem but also an "access denied" on image styles that had cropped up too.