
The primary restriction is the Land Clear clause of Rule 5:
An aircraft flying over a congested area of a city town or settlement shall not fly below such height as will permit, in the event of a power unit failure, the aircraft to land clear of the congested area.. This effectively limits flights over central London to twins and helicopters (which are exempt from the land clear rule). HOWEVER the big star-shaped restricted area (R160) applies Land Clear to helicopters.
As you can see, the eastern side of the city is inside the London City CTR (class D), and the west side is inside the Heathrow CTR (class A), the whole area ic under the London TMA (class A 2500ft +). If you can satisfy the land clear rule, your chances of a transit or SVFR transit are reasonably good, subject to traffic, your routing, the whim of the atco and so on.
Now, as for the BBMF (Battle of Britain Memorial Flight), they are military aircraft flown by military personel, and as such, they can go pretty much where they like....