The telephone exchange
Many UK exchanges are in old buildings due to origins as providing telephone services.
LINX in London is one of the main providers in the UK that interconnects networks, reducing the physical complexity and cost of connecting (peering) with other networks. An ISP would just need to have a connection to that exchange to allow it to peer with other networks.
Manchester is the second most important as it also has a physical fibre connection across the Atlantic.
Scotland has one in Edinburgh (IX Scotland). What can make interconnect exchanges successful is media providers deciding to use the exchange to store a copy of their content (e.g Netflix). Storing locally / caching reduces bandwidth usage across the Internet, and also adds resiliency in the event of network failures.
Juniper Networks is a popular provider of hardware to provide carrier network connections.
How do networks connect to each other?
Each ISP is allocated a unique numerical reference, and a system called BGP is used to allow physical devices (routers) to share route information by announcing what IP addresses it is responsible for.
Examples of how this works:
http://stat.ripe.net is a site that lets you see the peering that is occurring in near real time.