http://www.unitedutilities.com/sewers-and-drains-explained.aspx
United Utilities - “Sewers and Drains explained”
What United Utilities is responsible for – the big public sewers that take sewage and rainwater run-off to wastewater treatment works, the pipes from your property’s boundary that lead into the public sewer. Many of the shared drains where several property’s drains meet together before they join the public sewer – if you suspect that this is blocked, please contact us.
Severn Trent’s article on the same topic provides a useful diagram showing the extent of responsibilities:
http://www.stwater.co.uk/businesses/waste-water-services/sewers-and-drains/
You will see that the common drain joining the terraced houses is the water company’s responsibility (after the drain leaves the first house) . I think this covers your situation. This also supports gingejbee’s remarks.