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the Telewest box on the outside of your house will have a splitter that breaks the large cable into the smaller phone (stranded wire) and TV/broadband coaxial cables. Technically the cables into and out of that external box are still part of Telewest's installation and shouldn't b be messed with by the consumer, right up to the master phone socket, or the termination at the cable modem (CM) / Set Top Box (STB).
By the sounds of things someone has been quite adventurous in the wiring of your house, however, there is no way of knowing how much of that wiring is actually connected.
Cable connections for the STB (TV) will use the threaded connectors, unlike the standard terrestrial aerial that uses a push fit.
Potentially, if the wiring is connected, and the thread for the connector matches, your cable modem can probably use the wedge. Try it and see (if the modem lights are not steady, it's not gettting a signal suggesting wires are not connected somewhere).