HGiles: GOG's https server has been having problems since the winter sale. I wouldn't hold your breath waiting for it to get fixed.
I hope they fix the basic connection anytime soon, as no https is working at all. Right now, once I log out or change my IP, I'm gone. I'm pretty sure that logging in again doesn't work either. So if I don't answer anymore, don't hold your breath ;-)
It could still be a local problem on my side, but I didn't fumble on anything and out of a sudden half an hour ago, https was down.
EDIT: It's working again! Either it was some routing, DNS, HTTPS server or whatever hiccup or the Opera browser screwed something that was fixed be restarting it.