Should London implement jaywalking legislation in a similar at to the United States, Austria etc. Pedestrians stepping out wherever and whenever they wish is a risk to both themselves and other road users and it slows traffic flow.
In Austria you can only cross when a pedestrian crossing is green and cars, if no one is waiting can move across the crossing.
You would need to increase the number of crossings and introduce some form of zoning but could have a significant impact of correctly implemented.
PS: I don't drive, occasionally cycle but tend to walk most of the time.