In addition to @Keith_Beddoe 's advice you'll need the full aerial cable replaced & possibly the aerial itself and I'd suggest that you have a box fitted that the down cable goes into and an aerial fly lead from the Youview box to the new aerial socket. This way if the situation occurs again the water ingress will stop at the box and not get to the Youview box. The same can be achieved by using the above fly lead connected to the socket at the end of the down cable. Either way there'll be a barrier that'll prevent the water getting to the Youview box.
Hope that makes sense!
In truth why should BT replace the unit as it's not a component failure with the Youview box but a fault with your aerial. Maybe your home insurance should be the right approach.