I can't vouch for London but all non-essential businesses have been closed, those that are open limit the number of ppl allowed in-store at any one time and the police are actively discouraging ppl from travelling or wandering about unless they absolutely need to beyond a bit of daily exercise. Personally I've given up going to the local park for walks as there are too many ppl jogging all the time for my liking. I think there's a collective delusion doing the rounds that somehow if you're jogging you're either immune to the virus or incapable of spreading it. This virus isn't just spread by coughs, you're breathing out water particles all the time.
What does non-essential mean though? Everything in the US appears to be an essential business for example.
Also when did the non-essential thing start? As far as I remember people were still allowed to go to work if they couldn't work from home, at least when the lockdown started.
I mean, let's be real here. People shouldn't be out and about if you want to do a lockdown properly.
Imagine your lockdown situation like this. The only shops that are allowed to be open are grocery stores and drug stores. Only people that are allowed to go to work are people who work in those stores, or work in a medical facility. Only one person allowed outside the house once per day to grab groceries. That sort of lockdown. If you do a lockdown like this, I guarantee you new cases will drop after 2 or 3 weeks (assuming you are doing proper testing of course).
If you are doing some half-assed in-between lockdown, it's going to take longer before the peak, and the peak will be higher. You will certainly "flatten the curve" somewhat, but it's still going to be too high probably.
I really have no idea on what the situation is in the UK though as I don't live there, so maybe they are doing a pretty good job.