I've been too busy to have a play with this over the weekend, but I have just managed to get the site back up by taking the following steps:
1. Did a manual re-install
2. Deactivated all plugins by renaming the plugins folder to 'plugins.hold' via FTP access.
3. I could now log in to the admin panel. The WP install screen said that it had still to finish updating the database, but that is presumably because I'd just done the manual re-install?
4. Still the same blank screen with ATA 3.6.4. I rolled back the theme to 3.5.3, and everything is working again.
5. Reactivated plugins one by one, none of them caused any problems. It definitely seems to be theme-related.
I did contact my web host, and was told that the server doesn't have suhosin installed, and that as my site is in a sub-directory, there isn't a php.ini file that would affect me(?). I certainly don't have access to it anyway.
**edit** supplemental - Having gone back to ATA 3.5.3, If I try and preview the 3.64 theme in the admin panel, I get a preview of the blank screen!
So, I'm not much further on in solving the problem, but at least I've got things running again for the moment.