I'm glad you found it of some use. I altered the data download since the one you saw to be
https://api.coronavirus.data.gov.uk/v2/data?areaType=overview&metric=newPCRTestsByPublishDate&metric=newCasesBySpecimenDate&format=csv
Using cases by specimen date has removed the strange blip caused by them dumping in a batch of positives found down the back of the rest room sofa.
Using newPCRTestsByPublishDate rather than newTestsByPublishDate still leaves the ~0 .3% minimum but the data does ripple and smoothing to a 7 day average shows ~ 0.6% throughout July and August as you say backing up the suggestion that there was close to no infection then.
My original NHS England weekly file graphs and spreadsheets (I added under and over 60 lines and put some percentages in) show that deaths pretty much bottomed over that period with a lag of 2-3 weeks.
https://covid.digitalham.co.uk/age_morbidity.php
