Even with a PfCO, you’d need justification for the flight especially when in the heart of a major city… Just because you wanted to is not a good reason if anyone questions you.
Without a good reason, you will struggle getting permission to fly in the area, so good luck being insured.
Not to mention your pretty much bang on 5 miles from an airport.
My advice is that unless you want to be in the papers and on TV for all the wrong reasons, I’d not advise it! London never sleeps!