It'll Be Fine
The odd tosser aside, we're pretty good in the UK about working out how to avoid the worst of the tragedy of the commons - at least where we can apply some sort of hierarchy and deference. I think this'll be fine, will improve the pedestrian and cycling experience, and probably won't even make that much difference to overall journey times.
Non-automated cars will also find they are forced to follow the same behavior as the automated ones as the number of automated cars builds.
Will it be easy for all cultures? Not sure - having recently spent a Friday rush hour driving through Paris, towing a trailer, it does strike me that it will prove more difficult for some countries than others (that said, Parisian traffic junctions have, imho, led to the evolution of a particularly unhelpful style of driving).