forgive me if i missed something... if the computer reports "network cable unplugged" when there is a router between your pc and the modem, then surely the problem lies between the PC and the router ? The pc shouldnt report a network cable unplugged unless you power off or reboot your router... The modem falling over shouldnt cause that to happen... have you looked at the possibility that it could be a failing ethernet card ? You dont mention if you have tried replacing this.
Oh and quick tip.. firstly, this is an independant site, so talking to ntl in the first tense, doesnt sound quite right, and second, slating tech support isnt likely to get those here who do happen to work for NTL and hang out here, to help you.
just my words of advice.. i know how frustrating these things can be.