I was a Flying Crew Chief (FCC) on C-141's and KC-10's. Typically, with 141's we only flew if the aircraft was going to destinations that did not have enroute support for that airframe. For example, if a 141 was going from Dover to Ramstein, then no FCC needed. If going from Travis to Wake Island, then one would accompany. On KC-10's there are only two bases that have that airframe so we always flew with a minimum of 2 maintenance personnel and up to 4 dependent on the mission and aircraft configuration. 3 personnel was most common as many ground tasks required 3 personnel per the TO (refueling for instance). We flew under the status of MEGP (Mission Essential Ground Personnel) and did not have "flight status" per se although we were flight crew in every sense of the term. I finished my career with over 10,000 hours of flight time in 26 years.
Now particular to the duties, a Crew Chief would have knowledge of all acft systems, but generally not mastered on more than one or two. Jack of all trades so to speak. A CC would perform all routine maintenance and be involved with all other maintenance to include the direction of specialists, requests for additional maintenance personnel if hard broke somewhere, the keeping of aircraft forms and relaying maintenance status and need up to chain of command (up to major command HQ.)
As far as WWII specifically, I cannot say.