Great idea (Not)
This way it'll be able to track down which phone masts are giving us the virus.
So I'm in my house and somebody walks past the front window. Ping! and a light flashes on the console of the hollowed-out volcano.
Hey look Sarge. Red alert! Get out the riot gear. Ice cream van in the park has hundreds of people 'near' it.
* Remember GPS positioning isn't accurate to the nearest metre.*
There is no way 'near' can be interpreted as 'a close contact with a probability of transmission'.
The key to success is not space but time. If somebody tests as infectious then you want to trace contacts ASAP. Sooner if possible.
The ever so easy way is (a) record tracks with data remaining on personal mobile phone. This can be done as GPS etc, but more usefully as postcode-ish size zones and transport journeys and shops. I wouldn't expect the state of the NHS (National Hacking Service) to have the AI to work out you're on the 8:15 train from Biggleswade, but there is another solution. All busy areas have wi-fi hotspots. We can use those as beacons to tag locations including trains and buses.
(b) If needed, in the first instance this is a diary prompt for who was at your Dettol drinking party. The clue is in the phrase close acquaintance.
This might lead to a triage of people at further risk.
(c) Algorithmically we can upload a beacon/time history to a central server as step 1. (When infection confirmed.) Step 2: Now to alert people who might have been at the same place and time. "Hey" (subset of people known to frequent the busier hot spots regularly as a rough filter so as not to broadcast Carlisle stuff to Cornwall.) "mobiles, compare your tracks with our Lurgi Locus" Step 3 is a result of 'Coughing Charlie was in SHOPINGCENTREHOTSPOT at 11:43:12 to 11:55:35 2020:04:29 ' from which a human-target message can appear (info derived from (b) above) on the phone saying "An infectious male in his 50s, Blue Overcoat, receding hair, was in Sainsburys on Wednesday approximately between 11:30 and 12. He then may have caught the same bus as yourself."
False alarms will sink the system. The time/place scheme is going to be overflowing with irrelevant alerts. Great for fostering paranoia, but otherwise worse than useless. So the suggestions system outlined in (c) where there's real-world information on which people can judge is needed.