Assuming you have a recent version of InDesign and you're up-to-date with your subscription there may be a problem with the application.
Try resetting your InDesign Preferences:
Close InDesign. Then hold down Ctrl + Alt + Shift (Windows) or Cmd + Ctrl + Opt + Shift (Mac), and relaunch InDesign. Respond in the affirmative to the dialog asking if you really want to replace the preferences.
I doubt it is a problem with your Typekit.
When I search "Edwardian Script" on my Typekit, I see no results. However it still shows up in my InDesign as it is part of my system fonts folder.
You can try to download the Font from a third party website : http://fontsov.com/font/edwardianscriptitc54019.html
And then place it in the Default Fonts location, install it and restart Photoshop/InDesign.
I would say you also try to first sign out of your account from Creative Cloud and then sign back in. Still if you are unable to resolve your problem, you can follow the solutions mentioned here Troubleshoot font syncing problems in Typekit
It looks like it was installed with Microsoft Office 2007. How/why it left is a question for the Microsoft forum.
As Aman said, it is not part of the Typekit library, but you can license it from a variety of places. Just google "License Edwardian Script". Note that if you don't download/relocate your exact version, you will need to use Type > Find Font to update your files to the new version.