As Northern Italy and Madrid have proved, partial lockdowns just lead to many people trying to "escape" the area, go to second homes etc etc.
Also how will "London" be defined - just London boroughs? Is there really any different risk of somone in Surbiton (london borough) to Thames ditton (Surrey county council) just down the road.
The last clear stuff is the more confusion/lack of compliance, people thinking "it densest apply to me"