“A bun in the oven” means to expect a baby or become pregnant. It is a euphemism of becoming pregnant as literally an oven means a source of providing warmth.
The origin of “a bun in the oven” is stated to have been traced to, Cruel Sea, a novel by Nicholas Monsarrat published in 1951. There is a reference to pregnancy as a character Bennet says “I bet you left a bun in the oven” by which Bennet means pregnancy.
Examples in Sentences:
Example #1: It is mostly used as a euphemism such as “After various miscarriages, his wife was very happy to have a bun in the oven.”
Example #2: She has a bun in the oven after having passed her whole life without becoming pregnant.
Example #3: Both of them were happy beyond any description after having the news of their having a bun in the oven.