BuddyPress.org

Turns out it might be a bug after all, or at the very least an incompatibility.

If a plugin calls ‘get_template_directory()’ on any page other than the blog (ie if it runs on the admin panel) BuddyPress intercepts the call and rewrites the theme directory thus killing all your themes.

Thanks for all your help.

(I had Maintenance Mode on, which I activated when I upgraded.)