The "Use CURL" was the very first thing that I tried, but I got the same error.
In fact, it looks like our web services provider has suddenly changed the firewall settings on the server without informing me, and has blocked local access to port 443. I confirmed this by trying both wget and lynx from the server shell, and in both cases, attempts to connect to my own website failed, whereas connections to other websites succeeded.
So this may not have anything to do with the upgrade from 8.0 to 8.1. I'll be contacting our provider about this, and will ask them to restore local https:// access. I'll let you know.