For a long time I had to upload images using "Choose File" one by one. Then, a drag and drop feature appeared instead. It's now disappeared again, so how do I get it back and keep it there?

Is your uploader_jQuery plugin enabled?

Yes. I did try disabling it, restarting Windows, and then enabling it again. It might have worked the first time it did it, but not thereafter.

You should be seeing this once you selected an album or add a new one: https://www.zenphoto.org/screenshots/zenphoto-admin/upload-image-jquery-multifile.jpg.html

The area below allowes drag & drop.

Your screenshot is was I had for a while, but before that, and currently, it is this: https://rchsimagearchive.org.uk/index.php?album=Air-Transport#images-2

I've tried disabling it, restarting Windows, and then enabling it again, but to no effect.

I see that the tab Images (jQuery) is present. Click it and you get the page @acrylian showed you.
(you have to select an album or Create a new album first before you see the upload area)
If you only want to see the jQuery uploader tab, disable the uploader_http plugin.

I now understand the solution to my problem, but it isn't intuitive. It appears that when selecting "Upload Here" from the Admin Menu, you are taken to whichever upload screen you used last time, with the destination album pre-filled for you, but if you then change to the other upload screen, it is blank. It would be a lot better if it wasn't

Yes, the uploader used is remembered. Easy fix is to just enable one uploader plugin so you only have one page.