My sense of this kind of locution (and I hope my English cousins will forgive my presumption in speaking on their behalf) is that it is primarily a regal euphemistic mode. A royal personage should not be considered to have done something as "common" and "crude" (please note the quotation marks) as delivering a baby. It must be elevated to the level of "being delivered of," as if the process were somehow performed magically by the mother's royal status itself.

On a more generic and broadly applicable note, pregnancy and giving birth have long been referred to euphemistically even for non-royals. The reasons for this are recorded historically, and I won't go into them, save to say that (as reprehensible as we find these negative attitudes now) these natural and beautiful things have been regarded with shame, embarrassment, and worse. As one example, here's a very old medical term which even now is the abbreviation we still use, EDC, meaning anticipated date of birth. But what does it stand for? "Expected date of *confinement*." What a weird and subtly negative euphemism!