You have multiple variables in this problem. Whether the HLS Playback is the issue or Encrypted video is the issue, or Switching network itself is the issue.
So first,
1. Play a youtube video in WiFi and switch off the wifi router to see the video continue to playback even after the buffer. (Pick a longer video and enough playback to really make sure the buffer was over and is still playing. OR check the network logs to be 100% sure)
If that works fine, then we know network switching is not the issue.
2. If the Step 1 is passed, check a non encrypted video on your player and test the same. (Ex. "http://edu.infozen.cshls.lldns.net/infozen/edu/edu.m3u8")
3. If the step 2 is failed, it could be your player.
4. If the step 2 is passed, check whether you are using a standard aes encryption or any type of DRM which may stop streaming in such as change.
For all this, a network capture tool is a great way to proceed. It can tell you whether the requests are made or not. If made whether it is success or not.
(Charles Proxy and Fiddler are simple tools. You may use wireshark as well)