So, just to give more details:
For my first attempt, I tried to use the built-in File Manager in cPanel. It doesn't allow you to upload folders, so I did it piecemeal. Then when I started the install, it said there were no files to update, just some database changes. Once that was done, I visited the admin panel, and saw that the version was still listed as 3.3.3. I renamed the install folder as OLDinstall, to re-enable the board while I researched further. Then I discovered the topic pages weren't working, although the forum pages and home page still work.
For my second attempt, I'm not using the File Manager but an FTP client. It seems the one I'm used to, FileZilla, has been flagged as containing malware. The users on the forums are debating whether it's been falsely flagged or not, but my system is not letting me use it. So I downloaded CoreFTP, and am now uploading the files again. It's taking a looooong time.
More info as it arrives...
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Melkor and Ungoliant in need of some relationship counselling.