In the UK mil environment the requirement for engine air test following replacement or even adjustment is dictated by the individual Engineering Authority. Primary drivers are the performance of the engine at the extremes of the aircraft operating envelope, thus relight performance high and slow, slam from idle to max military power, fuel flows at altitude etc. Effectively things which cannot be confirmed on the ground. The main difference of course is the military engine is required to cycle rather more often per sortie and potentially needs to be operated closer to the limits than civ engines. I have known cases where early in a types life far more flight (and ground) testing was carried out than when the aircraft was more mature, reflecting increases in operating knowledge and commensurate increase in confidence regarding equipment performance and reliability. In answer to the question, in the UK mil it depends, but the decision is made by the engineers- this is flight safety we are talking about.