Not really convinced on the browser issue, sounds like Telewest passing the buck.
Might be worth trying an alternative browser to see whether that works before yo start deleting anything of the original configuration. Some of the guys on here recommend OPERA but I haven't tried it myself so I can't comment on how easy it is to install.
If you have managed to PING the Telewest host then there should not be a problem with your actual connection.
Have you checked the connection info inside IE6 to make sure nothing has changed from the original setup. If the problem PC is using the same connection type as you, I suggest you compare it to your machine, to save me typing the various steps you need.