What's a "significant fraction"? Reducing traffic by, say 70% and eliminating all the parking space that goes along with the millions of cars could massively improve city life. Air pollution, noise pollution, deaths and injuries in accidents, all those things would be dramatically reduced.
BTW, pedestrian zones exist in most cities. The stores there still get their goods delivered and emergency vehicles still have access. Turning most of a city in a pedestrian/cyclist zone doesn't mean that you can't have any car traffic at all.
I just looked out the window of our office and took a quick sample of the traffic. There's no bus route on our road, mind. Out of 10 vehicles, 2 looked private (no livery at least - one Mercedes S-class possibly chauffeured, the other could be Uber but wasn't a Prius so maybe not), 3 were black cabs, 3 were delivery vans and 2 were trade vans.
Eliminate 70% of that traffic, and something will break. Kill taxis and business will suffer. Kill trade vans and basic utilities will be out of action for longer. Kill deliveries and the pace of economic activity will slow down. Deliveries are the best candidate for reduction, but don't forget use of delivery vans is directly related to public transport use - people who use public transport can't carry lots of stuff around, or trek out to centralized pickup hubs, so they need delivery vans instead.
Hasn't London already done quite a bit to reduce private traffic in the city centre? Like you gotta pay 10 pounds to drive into the city center? That may explain why there's a smaller percentage of private vehicles, and quite possibly fewer vehicles being driven altogether.
BTW, pedestrian zones exist in most cities. The stores there still get their goods delivered and emergency vehicles still have access. Turning most of a city in a pedestrian/cyclist zone doesn't mean that you can't have any car traffic at all.