If there’s a one-step means of notifying Nightingale or Songbird that all of your media files have moved to a different location via the music players’ UIs, I didn’t see it, so I solved my problem with a short (~150-line) Python script [Python 3.6 (Windows, 64-bit)] that, with input via a few prompts, connects to Nightingale’s or Songbird’s email@example.com SQLite database (on Windows, in C:\Users\user\AppData\Roaming\Nightingale\Profiles\sth.default\db\) and changes the beginnings of Media_item content_urls.
The standard disclaimer applies. I give no warranties or guarantees. Use at your own risk. Make a copy of the SQLite db file before monkeying around with it.