Some time later Mrs Roberts is talking to a friend and says, I was very happy when the doctor told me that I had had a healthy baby.

  The conjunction that can be omitted after some common reporting verbs like say, tell, etc.:

I was very happy when the doctor said (that) I had had a healthy baby.

I was very happy when the doctor said to me (that) I had had a healthy baby.

I was very happy when the doctor told me (that) I had had a healty baby.