I used to work IT night shifts.
We were a NOC (Network operations centre) that monitored client networks. When something went down or was degraded (network was slow), we would troubleshoot and either call in a more advanced techie, call the telco, or call the site to physically check the device. Some nights it was quiet and our team (3 people) could sit back and chat, others is was a complete gong show from start to finish. Our customers were worldwide, so we would be dealing with Chinese, Japanese, Indian, Middle Eastern, South African and European customers, as well as any power outages in North/South America.
BT (British Telecomm) would likely have a simliar set up... maybe not monitoring, but taking calls from groups like ours. Almost all major telcos are 24/7. There were also any number of IT people "on-call" for page outs when things *really* hit the fan.
Data centres, server support, some application support groups all have a 24/7 team.
Let me know if you have any other questions. Nights shifts were a lot of fun.... as long as I didn't want a life!
Michelle