Servants having a baby - how shall the lordship react?

I assume you currently have the "butler" and "cook" set as full household members?

If the butler is a valued servant, I would have him retain his full status and remain a full member of the household. He and the cook could marry, but she could move out (no need for a divorce, just a household transfer) and raise the child in a separate household. However, she could be retained professionally by having the duke and duchess still hire her as a caterer regularly.