Base theme no longer recognized
I ran _system_rebuild_theme_data() but that didn't seem to improve the situation, so I checked the theme status in the system table; the base theme had been disabled. There lay the rub.

How or why the base theme became disabled remains a mystery. At the time this site was built, it was not required for a base theme to be enabled for the subtheme to access its functions; this changed, but I hadn't touched the theme in the month I've been working on this site, other than rebuilding the cache, so presumably it had been enabled before.

Since Omega was disabled, it was not supplying the right theme variables for our custom subtheme (though .tpl.php files did seem to get found), thus the broken display this morning.

Because Omega 4 is designed to be built and configured programmatically, perhaps, the base theme is hidden by default, and there was no way to enable it from the UI. I had to edit the .info file so that it was no longer hidden, at which point it showed up on /admin/appearance as usual.

If I had been able to use Drush on this server, this all would have been a non-issue. Many thanks to @Clive for his quick response and suggestions.