I don’t have an answer to this question, but I’ve gathered a few related stats that helped to get my own thinking into order.

The UK government is running two randomized assays of the population to monitor the coronavirus situation in England:

• REACT1 is a “PCR” test of approximately 150k people per month. This measures the number of people who currently have the virus.
• REACT2 is a “IgG antibody” test of approximately 150k people per month. This measures the number of people who have had the virus at some point in the past. It is unknown how long covid remains detectable via antibody test, but the REACT authors say: “most infected people mount an IgG antibody response detectable after 14-21 days although levels may start to wane after ~90 days”

By comparing REACT2 prevalence estimates to the fraction of the population with a confirmed case according to the government data dashboard, we can estimate that the true number of cases in the population is roughly 7-10x the number of diagnoses to date.

Start End N Sample Prevalence Cumulative Cases Cumulative Cases (% Pop) Sample Prevalence/Cumulative Cases (% Pop)
REACT2 Round 1 20th Jun 13th Jul 99908 5.96% 245000 0.44% 14.60
REACT2 Round 2 31st Jul 13th Aug 105829 4.83% 270000 0.48% 10.73
REACT2 Round 3 15th Sep 28th Sep 159367 4.38% 370000 0.66% 7.10

At the time of writing, 1053k people have tested positive, so conservatively 7M people have already had covid (i.e. 12% of the country). As a sanity check on this number, the REACT2 Round 1 report itself estimated 3.36M people as of the end of June.

Herd immunity refers to the situation where the reproduction rate, $$R$$, is less than 1 (and thus the number of infected people starts to shrink) because enough people are infected that the virus can’t find susceptible hosts faster than it is killed by the carrier’s immune system.

Assuming people mix homogenously, you need a fraction $$\geq 1-(1/R_0)$$ of the population to have got the virus to achieve this, where $$R_0$$ is a property of the virus that measures how easy it is to spread. Estimates place covid $$R_0$$ around 2.5-4, with 2.5 looking like the consensus of the BMJ for England. Thus the herd immunity threshold sits around 60%, i.e. 34M infected people. We would have to 5x the number of currently infected people to reach this level, probably increasing cumulative deaths by 5x in the process (i.e. incurring 187k new deaths).

However, some argue that under more realistic population-mixing assumptions, the threshold could be somewhat lower than this: perhaps as low as 10-20% of the population i.e. 5.6-11M people, which is a range that we have already reached.

The REACT1 reports use the prevalence numbers to estimate a nationwide R. This currently stands at 1.56 after, rebounding off a low of 0.57 achieved around the time that the most restrictive lockdown measures were released on May 10th:

Start End N Sample Prevalence R
REACT1 Round 1 1st May 1st Jun 120620 0.16% 0.57
REACT1 Round 2 19th Jun 7th Jul 159199 0.09% 0.6
REACT1 Round 3 24th Jul 11th Aug 162821 0.04% 1.3
REACT1 Round 4 20th Aug 8th Sep 154325 0.13% 1.7
REACT1 Round 5 18th Sep 5th Oct 174949 0.60% 1.2
REACT1 Round 6 (partial) 16th Oct 25th Oct 85971 1.28% 1.56

The R estimates here are convincingly higher than 1, so my feeling is that unfortunately the optimists who expect herd immunity to be with just 10-20% of the population infected seem incorrect at this point, and we would have to incur substantial numbers of new deaths to bring $$R$$ under 1 in the absence of measures such as the recently-announced lockdown.