Is the flat tire still in the vehicle (like in the trunk)? If so, it could still be picking up that sensor. Your spare most likely doesn't have a sensor at all, so it thinks your flat tire is still there because it's getting a signal from it. Eventually your flat tire will realize it's not spinning, stop transmitting and go into standby. I think that can take a couple hours.
Edit: I read your post again and realized that you already replaced the flat with a new tire. Are you sure the tire shop filled it up all the way? Use a tire pressure reader to see. No, it does not take several days to read the most current pressure from that TPMS sensor, it should see it within a minute or two of driving around, and it will display -- for that tire if it goes for more than a few minutes without seeing it.
Edit: I read your post again and realized that you already replaced the flat with a new tire. Are you sure the tire shop filled it up all the way? Use a tire pressure reader to see. No, it does not take several days to read the most current pressure from that TPMS sensor, it should see it within a minute or two of driving around, and it will display -- for that tire if it goes for more than a few minutes without seeing it.