You need to turn on two things.
Steps
- Open Settings > Alarms and switch Ignore Do Not Disturb on. This is the global permission.
- Open the alarm (or its group) and switch Ignore Do Not Disturb on.
- Tap Save.
Tips
- Both settings are required. If the global one is off, the alarm can't ring during Do Not Disturb.
- Combine this with a loud ringtone and a specific sender, so only important messages get through.