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Let's address the remote control issue first. Some STBs use the fairly common IR (RC5) signal. Many use a less common, and more troublesome to transmit through repeaters) IrDA. Any remote sender uint needs to be compatible with your type of remote signaller. See
http://www.redremote.co.uk/
Getting the TV signal to another room from your existing STB is the easy part. The most basic is to run a standard co-axial cable aerial from the STGB RF out (aerial out) to the next room and plug it to your other TV. Youare stuck with the channel being transmitted by the STB, on both TV in the main room and next door, unless you wander back the the STB with your remote, or use a remote control extender as above.
The more complex method is the use of video senders. These generally plug into the scart output of the STB, and send the TV signal wirelessly to the next room. Remote control is also handled wirelessly. It's the remote bit that generally is the incompatibility issue. Have a look at
Digisender
A cable splitter won't help you, as unless you get a second STB (from Telewest) you'd have nothing to decode the signals onto the second TV.