As it is pedestrianised, pedestrians have priority. That means you can't run them over if they get in the way. Even though it is pedestrianised it is still open to some motor vehicles, not just buses, and the local speed limit of 30 mph still applies. Stagecoach buses have a sensible, company imposed speed limit of ten mph, I am not sure what speeds other bus companies limit their drivers to but I used to think it would be the same.
Then yesterday I came out of the Tesco Metro you can see on the left just on the roundabout and looked to see if anything was coming. There was, a bus (not Stagecoach) some way back on the roundabout and I decided to cross as there was plenty of time. Well if the driver had been doing ten mph on the roundabout he wasn't by the time he got into Fleet Street. Probably double. I didn't have to jump for my life, just walk a lot faster, which at my age can be a bit of a strain. So much for pedestrian priority.