Apple Music doesn’t play music without VPN

It's almost certainly not a bug in iOS. If it were, there would be millions of people reporting this issue, and there are not. It's almost certainly a problem with one of the following:

1. Your carrier
2. Your iOS settings
3. Your cellular plan or settings (i.e. data restrictions, security features, or "data saver" features)
4. Something installed on your iPhone (i.e. VPN profile, DNS profile, other installed profile)

Therefore, a few more things to check on:

• Check: Settings > Apps > Music > Cellular Streaming > High Quality/High Efficiency (not NONE)
• Please check Settings > General > VPN & Data Management. What all do you have listed there? Any of the items there could be the issue because they can have a direct effect on network functionality. You may want to consider temporarily removing all of them while you are troubleshooting. Removing variables is a critical part of troubleshooting.
• Also, login to your cellular account on the web and look for any settings that offer enhanced security or "data savers" features from your carrier. If you find these, disable them.

Are you using Music while your iPhone is connected to a WiFi network at home, or possibly on a WiFi network that you do not control. It is possible that the manager of the network has blocked certain services from guest users of the network.

Have you checked that Music is enabled for Cellular data use:

Settings > Cellular > Tap "show all" under Cellular Data section > enable for Music