Wrist and core temperature

It takes 5 days or more for Apple Watch to measure and determine your baseline temperature. Please review the information in the Apple Support article I linked below,

Track your nightly wrist temperature changes with Apple Watch - Apple Support

>>About Wrist Temperature

Your wrist temperature is a measurement related to your body's temperature taken by your Apple Watch while you're asleep. Your body temperature naturally fluctuates and can vary each night due to your diet and exercise, alcohol consumption, sleep environment, or physiological factors such as menstrual cycles and illness. After about 5 nights, your Apple Watch will determine your baseline wrist temperature and look for nightly changes to it.<<

The Apple documentation indicates the temperature sensor is active while sleeping, and is useful for cycle tracking:

Track your nightly wrist temperature changes with Apple Watch - Apple Support

If that is not functioning, contact Apple Support, and ask them about this.

And alternatively, within 14 days of purchase directly from Apple, a full refund is available.