I suspect that "pregnant" was associated with animals, which is why the Queen might find it distasteful to use for people (even more so for royals!). This answer is a work in progress.
See the OED entry for "pregnant" (emphasis mine):
Etymology: < Middle French pregnant with child, pregnant (especially of an animal) (13th cent. in Old French; for earlier forms see note below; French prégnant ; now arch.), (of a word) full of meaning (a1585) and its etymon classical Latin praegnant-, praegnāns with child, pregnant, swollen, (as noun) pregnant woman, ...
I'm having trouble finding definitive statements to this effect, but see also the entry for "pregnant" in the 1828 Webster's Dictionary, especially sense (1):
PREG'NANT, adjective [Latin proegnans; supposed to be compounded of proe, before, and geno; Gr. to beget.]
Being with young, as a female; breeding; teeming.
Fruitful; fertile; impregnating; as pregnant streams
It makes no reference to human women, and "with young, as a female; breeding; teeming" does not strike me as phrases that would have been used for humans in the early 19th century.
Also compare the 2007 OED entry for the "with child" meaning (why it's so far down the page I don't understand) ...
II.3.a. Of a woman or other female mammal: having offspring developing in the uterus. †Also of the womb (obsolete). Frequently with with (the offspring), by (the male parent).
... with the 1989 definition, which does not reference women:
I. 1. That has conceived in the womb; with child or with young; gravid. Const. with, of (the offspring), by (the male parent).
Note that although the OED 2nd edition was published in 1989, the entry itself may have been written many years earlier (I could not find a date). If anyone has access to the 1st edition, I'd love to see it.
Importantly, the OED felt it necessary to add "of a woman" between 1989 and 2007.
However, this is definitely not universal. The concise Oxford dictionary of current English (1919) allows the usage for humans:
pregnant a. (Of woman or female animal) with child, gravid ...
Suggestions from other languages
As Sara Costa points out, the Portuguese word prenha - which is related to English "pregnant" - is used exclusively for animals (Wiktionary notes it as "derogatory" when used for women).
From Old Portuguese prenne (“pregnant”), from Vulgar Latin *praegnis (“pregnant”), from Latin praegnās (“pregnant”).
prenhe m, f (plural prenhes, comparable)
pregnant (of an animal)
(derogatory) pregnant (of a person)
This distinction is also present in French, where enceinte, from Latin incinctus is used for people (this is the word Lucy had to use back in the 60s), and plein(e) is used for animals, although plein(e) doesn't share a root with English "pregnant."
German, although not sharing the Latinate roots, does distinguish between human and animal pregnancies.