I have done something like this many years ago - so hopefully it will still work with the modern Virgin kit.
You can use the 2nd SCART output -designed for making external recordings- and plug into it a cheap SCART to RCA Output converter.
You then have 3 RCA sockets: composite video (yellow) and stereo audio (white and red). Then you ask a friendly satellite TV shop to make you some very long RCA leads to reach another room. Most TVs accept RCA input though you can get another RCA to SCART converter if the 2nd TV only has SCART.
Also, if your Virgin box is a V+, you can play hard drive recordings down this new long cable, by using the Copy To DVD function (which doesn't delete the recording afterwards). The person next to the Virgin box can still use all its other functions undisturbed.
Without using Copy To DVD, the 2nd TV will show what channels you are watching.
The 2nd TV user can also use a Remote Control Extender kit to control the Virgin box themselves.