Hello danielfromhilchenbach,
If you remove the Apple Watch from the charger when using nightstand mode, do alarms work successfully (when you're wearing the Apple Watch)?
Did you follow these steps to set up the alarm?
Resource: Add an alarm on Apple Watch - Apple Support
Set up Apple Watch as a nightstand clock with alarm
1. Open the Settings app
on your Apple Watch.
2. Go to General > Nightstand Mode, then turn on Nightstand Mode.
When you connect your Apple Watch to its charger with nightstand mode turned on, it displays charging status, the current time and date, and the time of any alarm you’ve set. To see the time, tap the display or lightly nudge your Apple Watch. Even nudging or tapping the table might work.

If you set an alarm using the Alarms app, your Apple Watch in nightstand mode will gently wake you with a unique alarm sound.
When the alarm sounds, press the side button to turn it off, or press the Digital Crown to snooze for another 9 minutes.

In case you did take these steps, which watchOS and iOS versions are you using? Have you tried recreating the alarm?
Let us know, take care.