Don't think you will find such a system is exclusive to BA. I know, for instance, that the B747 fleet of SIA has continuous monitoring of a very large number of systems parameters and performance and the information is automatically downloaded to Changi without any pilot input, (probably true of the A340 and B777 fleets also).
I am guessing that the majority of modern aircraft could have that capability and the major carriers will need it to stay ahead of the game and be able to forecast unscheduled maintenance, among other things.