Finally spent some time investigating this, and pretty much everything I wrote above is wrong...
The problem is an incompatibility between the download plug-in I'm using and the new version of the WordPress software the runs the site. I tried directly downloading the map file without going through Wordpress and everything was just fine. Took about 20 minutes to download the whole 12gb Map of the Pines HD file, which is comparable to what I've seen in the past.
I tried updating the download plug-in, but got an error message and (apparently) it has corrupted all the download pages on my site - when I go to them now, I only see a blank page!

Rather than waste any more time on this, I'm going to just get rid of the plug-in and create direct download links for all my maps. This will take some time however, so please bear with me.
I have already fixed the problem with the Map of the Pines HD download, so if you've been trying to download that map, the button on this page should work now:
https://boydsmaps.com/boyds-map-of-the-pines-hd-license-agreement/
Will post again when I've converted more of the map pages.