“How should a pregnant woman's skin care routine change?”

Skin reacts differently in different pregnant women. So, the answer to your question is, it depends. Some pregnant women are prone to acne and for some, the skin clears. Make sure there are no Retinol A products in your skin care regimen. Most benzoyl peroxide and salicylic acid can be used during pregnancy. See a dermatologist if needed.
Stay healthy!