I'm a bit on the fence about it. I read a thread here a while ago about 'a view from China', where they're doing exactly this. They were tracking people using two specific apps. If you had been in contact with someone who had it, you were whisked away into quarantine and your household were put into total lockdown.
On the one hand, it's a great idea on paper. So many of us come into contact with the virus and have no idea. But this totalitarian approach is not something that people in the Western world (maybe particularly Britain) will tolerate I don't think.
People don't like the idea we're being tracked, listened to, and if we 'sign away' these rights to privacy by signing up, who's to say they'll give them back when all this is over. In all fairness we would probably never know if they did or not.
I love the theory, but the reality is possibly not practical.
I saw a vaccine thread in the same vein as this yesterday. 'Would you get the vaccine if it was available right now'. So many people saying 'nope, everyone else can get it and risk it as it hasn't been tested'.