Yes, immigration. If they suspect she is entering to get free medical care, or to remain, or claim any benefit, or just have the kid in UK, they will stop her from entering. No, your kid has a right to citizenship in your country of citizenship no matter where it is born. You, on the other hand, derive no "rights" whatsoever by virtue of procreation.
Save the price of a plane ticket, etc, and skip the trip. It is expensive, embarrassing, and humiliating to be turned away at the border/immigration desk. Everybody will know what you and/or your wife is up to.