The reason the sites are not loading appears to be Vodafone's poor implementation of the Internet Watch Foundation web filter. UK ISPs are normally required to block or monitor access to specific URLs someone has flagged up to IWF. Any changes users try in Vodafone's Content Control interface have no effect on this.
When you access https://disqus.com or https://imgur.com for example, the OS finds the site's IP address from the modem and Vodafone's DNS servers. In this case, Vodafone's servers don't return the actual IP of disqus.com anymore, but rather their own IP 188.8.131.52. The browser then connects to that IP, requests the URL asked for, and presto, Vodafone has hijacked your connection and fulfilled their IWF duty. The only trouble is that Vodafone customers opening HTTPS links get an ugly error message instead and can't open anything on the site at all.
Your connection is not private
Attackers might be trying to steal your information from disqus.com (for example, passwords, messages, or credit cards). NET::ERR_CERT_COMMON_NAME_INVALID
This server could not prove that it is disqus.com; its security certificate is from contentcontrol.vodafone.co.uk. This may be caused by a misconfiguration or an attacker intercepting your connection.
Thanks to web certificates, the browser stops this intercept attempt. Vodafone's support suggests to ignore the message and reduce the browser's security, but that's not possible in all browsers. My daily browsing of Reddit nonsense through their mobile app forces imgur.com content over https, so all the images from there are blank. There goes part of the entertainment of a home broadband.
Further, if the error is ignored, the site seems to work, but the response headers contain this:
Via: 1.0 iwffilter.broadband.vodafone.co.uk (squid)
With the internet moving more and more to HTTPS, in many cases redirecting any HTTP access to HTTPS, this is going to become a major issue in the future. What can we do?