No, you can harvest a mature plant before it is an Elder version.
But, the plant can't have any Pests or Needs. You'd have to take care of those, first.
And, in the Test version at least, once you take care of any pests or needs, you had to leave your house area and come back in order to harvest. For some reason, it wouldn't automatically let you harvest after fixing the plant.
I often had to get pretty close to the plant to get the harvest prompt, too.
It means that you will be able to harvest it in 14 hours. Whether it is Elder or only at Mature level depends on the time the plant takes to grow between levels and what the re-harvest time is between harvesting. In test I noticed depending on how often I was able to login and check some plants would reach Elder before I could harvest them once, others I was able to harvest multiple times once they were mature before Elder state.