nRF24L01 - Measuring distance using time elapsed
Speed-of-light-delay measurements are not practical for the use-case you propose. As Majenko suggests in his comment, you'd need EXTREMELY accurate time measurement, as well as high speed radio-frequency signal processing. Any measurements you do through software will be far too inaccurate as a result of the software timing.
Apple's iBeacons (which are based on BLE) offer immediate, near, medium, and far distance readings that are based on signal strength. You should be able to achieve something similar with your own wristbands using BLE. The different distance thresholds would have some slop to them, but it should be possible to get decent results.