A teardown of the Apple Watch by iFixit reveals that the device indeed does contain the sensors needed to measure blood oxygen levels, not just heart rate. Apple reportedly seriously cut back the biometric functions of the watch because of doubts about the accuracy of the measurements.
The company believes that arm hair, sweat, and dirt could prevent the infrared light sensors on the back of the Watch from accurately measuring blood oxygen levels. Those sensors are supposed to send pulses of light down through the skin and measure the number of oxygen molecules that reflect the light back.
If the engineers at Apple figure some why of accounting for the interruption of those pulses of light by physical objects (hair, dirt, etc.) it’s possible the function could be turned on via a software update.