If this was modern Britain today I doubt the duke and duchess would do much about it, if it was say 1950s or so, they'd make them get married or fire the cook. In the Downton Abbey era, the cook would definitely loose her job, the butler perhaps be demoted or at least reprimanded, and he'd still be expected to marry her.
As you say you are playing in today's time, I think you have a great freedom in how you want to do it. The cook being pregnant is of course an inconvenience, since she'll be unable to perform her job to full satisfaction. They might need to hire a second cook and demote her to house maid, at least for the time being, until the child is old enough to take care of itself.
