Can a jet engine benefit from nitrous oxide boost?
For the combustor, I say the answer is no, because what is limiting the fuel rate, isn't the amount of oxygen (gas turbine combustors run very lean), but the ability of the turbine materials to withstand a higher gas temperature entering the turbine. Hence, to temporarily increase engine thust, water injection into the combustor was used on early jet engines, to increase thrust during takeoff. 

This sounds quite counter intuitive, but the explanation is based on thermodynamics. Naturally, the water evaporates into steam, which then gets heated to the temperature of the combustor exit temperature. This evaporation and heating process of the water and then steam, is drawn from the heat of the combustor, and so reduces the combustor temperature (for the same fuel flow). However, the engine does not need to add more fuel to make up for this cooling, at the same level of thrust, because the energy added to the steam is still available in the turbine, to be released as the combustion gas / steam mixture flows through the turbine, expands and cools. As it cools, it gives up this energy, which is effectively used to power the turbine. The remaining energy is then converted to thrust, as the flow expands through the final propelling nozzle. But now, because the turbine temperature is lower for the same thrust level, the fuel flow can be increased until the maximum turbine inlet temperature is reached, and so now providing more thrust. 

So, rather than inject N2O into the main combustor, inject water. 

For the afterburner (AB) however, the case is quite different, and I believe a performance benefit could well exist. AB combustion runs much closer to an equivalence ratio of 1 (about 0.96), i.e. burning almost 100% of the fuel level that can occur with complete combustion of all the available air. And unlike the main combustor, there are no real material temperature limits (cool bypass air exits along the inside of the AB duct surface). But, the question is, do you really want to do this, operationally? Is it better to use some of the planes tank capacity for some N2O, or just entirely for fuel? I suspect operationally it is better to be able to use the AB for longer, at a slightly lower thrust level, than shorter, for a higher thrust. I don't believe military aircraft use AB to out run missiles, so pure thrust isn't the only aim. But that's another (next) question....