Yes, you've basically got it right. The router will connect by ethernet cable from its WAN port to the Telewest Cable Modem.
Both PCs then connect to the router, either with an ethernet cable or "wirelessly" so, at the moment, the new PC will be able to connect by ethernet cable and will require a "Wireless Network Adapter" installed if you want to use that method instead. The old PC will need to have either a "wired" or a "wireless" network adapter installed so that it can connect to the router.
<<< I have just bought a new computer which i want to connect to the internet wirelessly via my old computer >>>
Being pedantic, the new computer doesn't connect via the old one. The *router* connects to the internet and both computers get their internet access over the Local Area Network from the router. Each has internet access totally independently from the other.