I'm an Avionics licenced chap and in our company i'm allowed to run all the types I hold to ground idle (after a 2 day engine run course). 2 in the flight deck at all times during the run and no unlicenced eng's are allowed to run engines.
APU's are a different matter, anyone with approval on type can run APU's in our company and as all the types we have now are glass cockpit you can't monitor the APU until it's up and running if there's no ground power. Once it's started they all have auto shutdown and no requirement for constant monitoring.
From what i've heard BA at Gatwick allow properly trained Eng/AF eng's to taxi when required??