The number of tests performed every day has increased from 15,000/day in April to 200,000/day today.

So when the number of positive tests today is the same as the number of positive tests in March, what does this mean? Can we estimate the true number?

The biggest clue is the ONS Covid Infection Survey pilot (yellow line). They send tests to people at random, and see how many come back positive. About 1 in 1,000 come back positive, which suggests the true number of cases is about 10 times the number of positive tests.

Can we estimate the numbers before May?

As a back-of-the-envelope guess, perhaps the proportion of the country which is infected is one hundred times lower than the proportion of positive tests. (ie: if 5% of people who are tested have coronavirus, then we guess that 0.05% of the general population are also infected). This gives a number (orange line) which is lower than the ONS survey, but looks vaguely plausible.

This guess would suggest the number of infections today are 10 times lower than the number of infections during the peak in March -- even if the number of positive test results (blue line) are about the same.

Here's my .ods spreadsheet with rough working out.

If you want to see someone do a much better job than me at making sense of the data, I recommend this map, made by @russss.

One final side note about the annual trends for seasonal flu, which I has expected to predict the second wave. The key time for this are normally from November/December to February/March. I wasn't expecting the numbers to get worse this early.