The oil pump is driven by the crank shaft not the chain. The pump is in the timing chain cover and the crank gose through the pump. The crank shaft has flat spots that drive the pump. If the chain breaks, the cam shaft stops.
If the pump were driven by the chain or the cam it wouldn't make much difference. If the chain breaks, the valves don't open or close, the engine becomes quite inefficient and the engine will stop. The need for an oil pump quickly diminishes when the engine stops. In fact, the oil pump on my Viper stops every time the engine stops.
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