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The easiest and most reliable way will be to run a new coaxial cable from the rooftop aerial to the TV point in you living room, thus bypassing whatever was done for the Telewest Installation.
An alternative (so you don't have to climb the roof) would be to cut the coaxial cable just before it enters the external Telewest box, make a reliable joint and attach a new length of coaxial cable between the joint and the living room wall point.
Of course with either of these options routing the new cable may be a challenge, getting it through walls.
The connector in the box in your living room will be a threaded "satelite" style fitting. It is possible to get adaptors that convert that to a standard fitting. Whether the existing circuit through the Telewest box could then be used, with a satelite to TV coaxial adaptor I don't know. Without a standard Telewest STB there may be too much interference.