Oh my God, I don't know what to say!
The point of lockdown is to get the situation sufficiently under control that the government can then carry on with testing, tracing and quarantine. Not give it a rest and then wait for another lockdown!
You realize that many other countries have got this situation under control without any lockdown at all?
Test and trace. Quarantine mild and asymptomatic people quickly in hotels. Quarantine all new airport arrivals in hotels. Everyone wears masks outside the home unless they are under 3 or have a genuine medical or psychological conditions preventing this. Carry on "stuff" (schools, businesses, restaurants, travel) but in a more distanced manner with lots of hand hygiene, temp checks, masks, spacing stuff out, moving operations outside (which will slow any spread of the virus, meaning that if you get a cluster, it can then be crushed quickly through contact-tracing and quarantine measures).
The virus is showing signs of struggling in warm weather. The number of cases in London has gone down. This is the golden opportunity to get this thing under control now. You can then carry on with "modified normal" until there is a vaccine. That's what they are doing in HK, Taiwan, and a whole bunch of other countries.