London traffic is nothing to fear in my opinion. As a child I was taught to stop, look, and listen before you cross the street. Cross walks give us a very welcome added measure of safety but it is our wise use of our own senses that truly keep us safe.
The lesson has served me well. Try it.
I watch cars, trucks and tour buses drive through the crosswalk at least once a week while i'm in the process of crossing the street at a crosswalk/stop lights with walk signal, often 5 seconds or more after the light has changed.