Marc,
I had a similar problem with my BTInternet ADSL broadband. The always on connection wasn\'t always on, on one day I lost the connection about 10 times and logged a help desk call each time. After these persistent calls to the help desk my fault got escalated to the highest level. Eventually the wiring within the house was replaced (their engineer gave me a roll of wire which I installed and they connected up). They also replaced a card at the exchange and replaced the Alcatel fish modem. They did all three fixes simultaneously so we never found out what the actual problem was, but I got a reliable internet connection out of it.
I also had a need for a SOHO network and used MS ICS (internet connection sharing) with various levels of success (sometimes it worked and sometimes it didn\'t !). My setup was an internet connection via the Alcatel fish USB modem to the main PC and networking via 8 port hub with ICS running on the main PC. However, as mentioned ICS was not always reliable, so once I had established a reliable ADSL connection, I upgraded the modem to a D-Link 504 Router (this is an ADSL modem with a built in 4 port hub, all configurable through a browser interface), this does exactly what is says on the box, is very reliable and any connection problems I have had, are due to BT\'s servers. I have also piggy backed an 8 port hub to the router to extend the network past 4 connections.
So in your case, I would connect the Telewest modem to the main PC and establish a reliable connection - make a nuisance of your self with their help desk until it is sorted. As regard the networking side, it is not their responsibility (why should it be), so if in the interim period, you need to run a SOHO network, connect a hub to the main PC (you may need a second network card) and use ICS for connecting the networked PCs to the internet. Once you have established a reliable internet connection, get yourself an all in one broadband modem/router (in my case, the D-Link is very good, but there are others available) and then connect all PCs via the modem/router. Once it is all working, you can get rid of the second network card and ICS, and connection to the internet does not depend on the main PC being switched on.
You may also be able to connect your existing modem to your router/hub, I have no experience of that type of setup, so I can\'t help you there.