I would suggest you power everything down then connect the problem laptop directly to the cable modem. Power up the modem and wait until it has booted fully and is online, then boot the laptop. Check that it has its IP address from the ISP (NTL/Telewest?) and whether it can access web pages, email servers etc.
You haven't mentioned whether or not File/Printer sharing has also been a problem with this particular laptop, which might indicate a TCP/IP issue.
Something to try is this little WinSockFix application if it is a TCP/IP problem click here