It seems that all customers of Virgin Media are being assigned the IP address 220.127.116.11 when they edit Wikipedia. Similarly, all customers of Be Unlimited are being assigned the IP address 18.104.22.168.
This produces all the usual problems associated with shared IP addresses on a much exaggerated scale.
I strongly suspect the problem lies with MediaWiki. That the same thing has happened with both these major ISPs at the same time suggests that the error does not lie with their configuration. Additionally, customers of these ISPs are not all on a shared IP address at all. My IP address is currently 22.214.171.124, as far as everyone except Wikimedia is concerned, yet I am affected by autoblocks and direct blocks on 126.96.36.199. Other users are reporting the same thing.