Wow, Steve. That's a brilliant idea. All you need is ..... a ladder, a couple of big lads, a spare SHF link transmitter, a digital feed of the studio output available on the roof to connect to the SHF transmitter, the ability to rig a line-of-sight link, some communications kit to allow you to talk to the BT Tower while you align the circuit, a couple of spare bods atop the BT Tower to align one of their dishes, and - bingo! you're in business. How long do you think that would take, Steve?
I think the nearest thing to anything like that happening was the 1981 Royal Wedding. I think the SHF receive point on top of Millbank Tower that Thames were using to get the St Paul's OB 'cluster' back to Euston failed, and someone from LWT went to the top of their tower, and manually steered one of their dishes and managed to receive the signal, sending it back to Euston by line from there ?
If you draw a line between St Paul's and Millbank Tower, it does come pretty close to LWT's tower, so a plausible story. While all that was going on, Thames simply took the BBC's clean feed. (Both broadcasters had available each other's output for this sort of emergency occurrence)
Meanwhile, in other parts of the UK