Okay, prediction.
The reason that Scarlet doesn't remember her execution, and the reason that she recognized Constance back then, is that, from her perspective, it hasn't happened yet. She's not a ghost. She's a time traveler. The story will end when the two of them triumph over the conspiracy, after which Scarlet returns to the past to fulfill her destiny, having (probably) destroyed her enemies and (hopefully) become a better person in the process.