That's an interesting article and yet another take on the situation that involves widespread testing and contact tracing. I don't really understand why UK testing is so restricted and can only speculate that there is lack of laboratory capacity or some other infrastructure reason.
12 PHE labs + contracted out. One the reasons that the media keep on taking UK 'average' numbers per day or quoting the total number to date is that it's extremely variable, sometimes the English bit of the UK does a lot of tests in a day, sometimes hardly any. It's clearly done in batches and a lot of driving around done.
The UK and Italy's testing is really weird. It started off about the same level and running parallel for a time. Then about ten days ago Italy started massively ramping it up. For how far behind Italy we are in virus progression, the UK was actually doing an all right number of tests in the early stages (and Italy pretty bad to be honest). This advantage has been wasted as it's not ramped up as it needs. If the stuff PHE had said was true we'd not be far behind Italy's testing regime until 3 days ago and places like London a week behind and testing regime for non-hotspots doing a lot better.
The BBC's Click edition on bbc news has a lot of stuff on testing this week.