Huw,
Got into your site, but couldn't log in with your Facebook account.. they wanted the answer to your secret question.
Anyways, I did the following things, which likely won't fix the problem, but will hopefully give more information to localize the issue:
1) We changed some code to re-arrange and perform a better check where we think the white page/PHP error was coming from - You hopefully won't get a blank page any more
2) We added multiple Joomla system messages for more status during the login process. Nothing should be intrusive/bad for real users on the site, but there are additional error messages now being output. When you try to login again, please let us know any of the red/blue system messages you get.
3) Your current front-end template was not showing system messages, so I added the following tag to your template directly above where the component is displayed. This lets the red/blue system messages show.. you may see more of them now as any extension, as well as Joomla, will use them for status messages:
<jdoc:include type="message />
That shouldn't be a problem to have, and is normal, but you can remove it after you get the status messages for us.
Please let us know if this improves the situation or gives us more details to narrow things down,
Alex