Jameseh

You should get shielded Cat6e instead, may as well get the best when your rewiring. Virgin Media cabling I think they require use of their own cables should you join, so you might be better off ringing up and seeing if they'll run the cables for you (at a hefty cost I imagine though). BT uses a mixture of ethernet and standard coax so that should be fine with the other two wires.

You might need to use some signal boosters along the way for the terrestrial if your planning on using one aerial to four rooms.

Good luck.