Re: Last month's solution?
Let's try somewhere in the middle. Don't just tell the traced contacts to lock themselves away. Assume that some, possibly many, will be false positives.
Treat the tracing as simply a presumptive test, i.e., one that has to be backed up with a better test. Only require the contacts to self-isolate in the first instance until a test result is available and then continue or not based on that.
Look to get quick turn-round. Instead of the mega-hubs where the test processing starts next morning distribute the processing so that it can be carried out closer to the subjects.
Look into tests that are supposed to give quick results on inexpensive equipment. Even if they also give false positives (but not false negatives) they can act as a second level test before sending samples for a more definitive test. But testing contacts is key, otherwise when the complaints start coming in in a few weeks time the whole thing will blow up in HMG's face.
If that leaves any spare capacity start looking at random testing. If there are asymptomatic cases giving rise to contacts relying on symptomatic cases only will take forever.