Crick was trained as a physicist, and was working on his Ph.D. at the time. Watson was educated as a biologist and already had his Ph.D. (this was back in the day when the big divide in university biology research was between zoology and botany, not molecular/cell biology and organismal/evolutionary biology).
I think there's a two or three year limit on the Nobel Prize being awarded to dead people. So if it had been a year or two earlier, Franklin should have been on that list. Not sure about this, though.