Cannot access login page a second time (White screen)

I’m using a Child Theme with TwentyFourteen and all’s been great for months until someone complained that they were unable to log in.
We checked the site and discovered that a bookmarked link (on the browser) to wp-login.php seemed to work fine, but the html signup links on our web pages resulted in a white screen.
Trying to access any admin page seemed to bring the same result (signup etc).

1. Deactivated all plugins – no difference.
2. Activated TwentyFourteen theme – no difference.
3. Re-named the plugins folder using FTP – no difference.
4. Downloaded a fresh version of WordPress, deleted all wp files and folders in my “public” folder EXCEPT wp-config.php and the folder wp-content, then uploaded fresh files – no difference.

For some reason, the fault seems to have evolved slightly.
On starting Safari, Firefox or Chrome, clicking one of the login links works OK and our members can view the content of our protected pages but if you log out and then try to log back in again, the white screen appears every time. Sometimes you can use the “back” button and try again, but at other times, the browser seems to freeze.

The website is here: http://www.workingsheepdog.co.uk/
Any suggestions would be very welcome.
Andy