Is there a conceivable reason that NASA or any equivalent space agency (ESA, Roscosmos, etc.) would need to send non-astronauts into space? It seems that for almost any mission, astronauts could be instructed on how to complete the necessary task beforehand or remotely. If not, machinery could also be operated remotely. Lastly, for planned missions (e.g. Mars), civilians with requisite credentials (e.g. Geology PhD) could be selected for astronaut training well ahead of time. In what situation would a non-astronaut of any expertise (or lack thereof) be required to personally accompany a mission.
The time period to consider is near past to near future, implying that the only people trained to go to space are astronauts and the odd paying tourist. The technology level to consider spans the same time period (think shuttle era to ISS to Deep Space Gateway/Lunar/Martian missions. Any length of mission preparation up to the actual length of astronaut training (~2 years) is acceptable.
You can invent any wild scenario to justify this (obviously "all the astronauts just got pink eye and the Hubble needs a bath" is a bit out of left field but hey go wild). If multiple valid scenarios are proposed, the most realistic/probable one will be the correct one. More weight will be given to the need to send non-astronauts than the want (e.g. sending scientists to do a sciencey thing rather than sending politicians for a P.R. opportunity).