Even in the exosphere, at 1800 deg centigrade, the only components of the Earth's atmosphere that can currently attain escape velocity are hydrogen (3 kg per sec), and and helium (next to nothing). The notion that the Earth's atmosphere would be stripped away by solar wind but for the protective magnetic shield, is a myth. So there is no 'characteristic time of loss of the Earth's atmosphere' in the sense of slow decay. There is certainty that, as the sun approaches the red giant stage, several billion years from now, there will be catastrophic heating of the atmosphere such that oxygen and nitrogen will be stripped away, leaving Earth to overheat in runaway carbon-dioxide. Unless, that is, if gravitational mayhem, caused by galactic collision with Andromeda, flings Earth substantially further from the sun.
Oxygen, nitrogen and inert gasses can escape from bodies the size of the moon, Venus and Mars because the thermal process, 'Jeans Escape', in less massive bodies incurs substantially lower escape velocities.