"Pregnant" as a taboo word
Solution 1:
I would guess that the word "pregnant" focuses on the state of the woman's body whereas a euphemism focuses (or at least pretends to focus) on the anticipation of a child.
Referencing a bodily state or function typically is taboo or disfavored.
Solution 2:
Although Etymonline lists pregnant as "taboo" until 1950, the OED doesn't make any such claim and lists many uses dating from 1425. This includes many medical uses of the word, but also includes notable literary uses:
1667 Milton Paradise Lost ii. 779 My womb Pregnant by thee, and now excessive grown Prodigious motion felt.
1914 T. S. Eliot Let. 31 Dec. (1988) 74 I hate university towns and university people, who are the same everywhere, with pregnant wives, sprawling children,..and hideous pictures on the wall.
If the word was taboo, Milton and Eliot didn't seem to think so.
On the other hand, Google NGrams shows that the usage of "pregnant" did increase after 1950:
This seems to indicate that, while the word "pregnant" wasn't strictly taboo in the way that profanity is, it may have been disfavored.
Note: the OED link may not work unless you have access through an academic institution.
Solution 3:
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.
English
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):
Pregnant
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.
Solution 4:
I don't have more to go on other then what I was taught, so I'm sorry for the lack of reference, but here we go anyway.
Pregnant is a medical condition. It is a state, with "treatments" and so on. You get there by having sex. You stop being pregnant by giving birth (or other outcomes). It's is an exact state relating to ones health. To say Sue is pregnant, is to say that Sue engaged in sex, the man orgasmed, she was fertile, "they" conceived, and now Sue in the process of gestation. Sue will one day spread her legs expose her vagina, and excrete a child.
Medical conditions like that are private and should not be discussed openly in public.
Saying that Sue is "in the family way" is very different. That means that Sue is buying kid clothes, decorating rooms, finding a sitter, saving money, planning for a college fund, picking out a doctor - and yes, somewhere in there, she probably has to get pregnant, but that's not important. What's important is Sue is making "motherly decisions". This is a vague term that can mean many different things. So it's "safer" to talk about.
Talking about pregnancy might as well be talking about sex and childbirth. Talking about a new expected family member leave room to be talking about pink or blue curtains.
Furthermore, with pregnancy in general, there is a very strong "don't jinx it" vibe. Things like not naming a child until after birth, not knowing the gender, not announcing until later on, etc all help keep the word "pregnancy" a bit more vulgar then the alternatives. It's OK to talk about having children so long as your not talking about a specific person or process.
Lastly, and most odd, is the carry over stigma of being pregnant. Though I will not understand it, never is a woman treated less like a woman then when she is doing the most womanly thing possible. This has a long history of being true. So if you can talk about raising children without discussing the "weakness" of being pregnant....