The Tosefta (Taanit 2:12) cited in part in the Bavli (Pesachim 54b) says that pregnant and nursing women must fast on Tisha Bav (in contrast to other minor fast days). No distinction is made for a postponed fast. Such a ruling is uniformly accepted and documented in all the classical codes, including the Mishneh Torah (Taaniyot 5:10) and the Shulchan Arukh (OC 554:5).
In general the laws of the postponed fast are exactly the same as any ordinary Tisha Bav. There are two Talmudic source which could indicate any degree of leniency for a postponed fast. The first (Megillah 5b) indicates that Rebbe for some reason wanted to entirely skip Tisha Bav on years when it was pushed off, but that the Sages did not accept that position. The second (Eruvin 41a) tells of a certain family who had an annual Temple-related celebration on 10 Av who wouldn't complete the fast when it was pushed off to 10 Av. The Mordechai (Moed 630) quotes an opinion that the same should apply to the parties of a circumcision (father/Mohel/Sandek) occurring on a postponed fast on 10 Av since that day is a personal holiday for them. The Shulchan Arukh (OC 559:9) rules accordingly that the parties of a circumcision can break their fast after Mincha on 10 Av.
In the last few hundred years some rabbis have found room for leniency for pregnant women based on this.
Rav Ovadiah Yosef is the first person I'm aware of in history to argue that pregnant women are entirely exempt from fasting on a postponed Tisha Bav. He argued that since on an ordinary minor fast day pregnant women are exempt but parties of a circumcision are obligated, in this case where parties of a circumcision are exempt all the more so must pregnant women be exempt. (See Yabia Omer OC 5:40)
There are many problems with his argument. First, many Rishonim (including the Rambam and Rif) do not cite the Talmud in Eruvin about personal holidays overriding a postponed Tisha Bav, and indeed many Acharonim dispute that the Halakha follows the view brought in the Shulchan Arukh (see Magen Avraham and Shaarei Keneset Hagedola about the custom in Turkey, Pe'ulath Tsaddiq 3:147 about Yemen, Arukh haShulchan about Eastern Europe, that "we have neither seen nor heard anyone act on this [and break their fast for a circumcision]... such is the custom and one should not change it").
Second, even those who are lenient for a circumcision generally limit it to after Mincha when one has already mostly fasted (cf. Shaarei Teshuva OC 559 sk 7). Only the Chemed Moshe (562:1) allowed having the circumcision earlier in the day and breaking the fast then. (Regarding a circumcision the Magen Avraham quotes two opinions whether it can happen at 6.5 hours or 9.5 hours into the day, and recommends being strict, though many are lenient.) Accordingly, pregnant women should only be allowed to eat later in the day. (R. Yosef relies on the minority opinion of the Chemed Moshe that the circumcision can even take place in the morning, and seemingly further extends it by arguing that circumcisions must take place during the daytime, but the pregnant women is already pregnant at night.)
Finally, the argument is predicated on a suspicious Kal Vahomer: a postponed Tisha Bav is more lenient than an ordinary fast because of laws of circumcision, so it must be as lenient as an ordinary fast regarding pregnant women. This is not obvious for two reasons. First, even a postponed Tisha Bav is significantly more strict than an ordinary fast since it still starts at night, has the 4 other afflictions, prohibition on Torah study, etc. How can you say it must be as lenient regarding pregnant women? Second, many Acharonim (Maharash Halevi OC 2, Beit Meir 559, Avnei Nezer 1:427,429, Eshel Avraham (Buchach) 550:1) actually make the exact opposite Kal Vahomer: on a postponed Tisha Bav women are obligated to fast while they are exempt on ordinary fasts, so parties of a circumcision who are exempt on a postponed Tisha Bav must also be exempt from ordinary fasts.
It's true that R. Yosef provides a couple Rishonim who hold that parties of a circumcision are obligated to fast on ordinary fasts, rejecting that conclusion, but we see that all those Acharonim took it as obvious that pregnant women are obligated to fast on a postponed Tisha Bav. Perhaps those Rishonim just didn't hold like the Mordechai that a circumcision overrides a postponed fast so we can't use them to compare to our case? Perhaps what we should be learning from those Rishonim is that Kal Vahomer shouldn't be used here, especially since the reasons for exempting pregnant women and parties to a circumcision don't seem to be the same (health and/or fundamental optional nature of the fast vs. personal holiday pushing off mourning).
(Looking for a moment at the idea that the permission for pregnant women to not fast on the three minor fasts is because those fasts are fundamentally optional and the community never accepted that pregnant women should need to fast, Divrei Malkiel (3:26) in a responsum during the sixth cholera epidemic ruled that a city at high risk of cholera did not need to fast on Tisha Bav because of the great danger. He bolsters his position by noting that a postponed Tisha Bav may be considered more lenient since he argues the Second Temple was burned entirely on 9 Av (unlike the First Temple which mostly burned on 10 Av) and therefore a postponed Tisha Bav reverts to fundamentally optional since it is no longer the date of multiple tragedies (cf. RH 18b). He also notes some Rishonim to the Talmud in Megillah interpret Rebbe's wanting to skip a postponed Tisha Bav as following a view that Tisha Bav is also part of the "fundamentally optional" fasts. According to this pregnant women could be exempt like on ordinary fasts, but 1) it's not clear he ever relied on this without the overarching concern for cholera, 2) there's no reason to think the will of the community was ever to exempt pregnant women from a postponed fast, and 3) it's not clear we rule like those Rishonim in Megillah, or that the Sages, who we follow, agreed with Rebbe about this point.)
R. Yosef indeed in a later responsum (Yechavveh Daat 3:40) backed off a bit and recommended pregnant women try to fast until the afternoon.
A somewhat similar argument was presented earlier in R. Yaakov Reischer's Shevut Yaakov (3:37), writing about Tisha Bav 1721. This source was quoted by R. Akiva Eiger and then in the Beiur Halakha (not the Mishna Berura, perhaps because he thought it more of an oddity / סניף להקל than mainstream Halakha) so it has received some recent prominence, but it's worth seeing the case inside.
אחד היה לו חולי שאין בו סכנה כ"כ וקיבל עליו התענית והתענה עד אחר חצות לזמן מנחה גדולה ונחלש עד שצוה הרופא שלא יתענה יותר והורה להם המורה שיתפלל מנחה גדולה ואח"כ יאכל ולפי שהתפלל תפלת מנחה ביחידות שכח מלהניח תפילין כדינו ושאל השואל מה תקנתו.
תשובה: הנה בוודאי מה שהורה המורה שיתפלל מנחה גדולה יפה עשה כיון שנדחה ט"ב על יום א' א"צ להשלים אפילו בחולי קצת וכן נהגתי להורות ביולדות תוך ל או במעוברות ומיחוש קצת דהא אפילו משום כבוד המילה הקילו בהנ"ל כמבואר באו"ח סימן תקנ"ט סעיף ט' כל שכן בחשש חולי...
Someone who was sick without too much danger and accepted to fast and fasted until after noon at the time for Mincha, and he became weak until the doctor ordered him not to fast further, and the rabbi ruled that he should pray Mincha first and then eat, and since he prayed Mincha alone he forgot to don his Tefillin and he asked how to fix the situation.
Answer: Certainly that which the rabbi ruled to pray Mincha first was correct, for since Tisha Bav was postponed to Sunday, he does not need to complete the fast even for a smaller illness, and so I have been accustomed to rule to women within 30 days of giving birth or pregnant women who feel unwell, for even for a circumcision are we lenient, all the more so with a risk of illness....
We see the level of sickness that he used to permit breaking the fast was such that the man, after being ordered by a doctor to stop fasting, hurried to pray Mincha even without his Tefillin, and that insisting on waiting till after actually praying Mincha was essential. This isn't quite a blanket ruling that all pregnant women categorically are exempt in the afternoon.
So you have this one source from the 18th century which allows for a somewhat weaker standard of illness to exempt from fasting, which gets quoted a couple times after that, and at the same time you have all the above-mentioned sources (Avnei Nezer, etc.), both before and after that source, who explicitly assumed otherwise or didn't cite any leniency for this relatively common situation (about 1/4 years). Even quite recent works like the Arukh haShulchan, who certainly saw R. Akiva Eiger's citation, make no mention at all of any leniency for pregnant women on a postponed Tisha Bav. It doesn't seem like leniency on this issue was at all widespread historically.
R. Shlomo Luria (Shu"t Maharshal 53, the earliest explicit source on the matter that I've found) ruled that even a woman who gave birth on Saturday 2 Av must fast on Sunday 10 Av since she is past her (OC 617:4) seven day recovery period. Even though the case was explicitly where the fast was postponed, he, by omitting any mention of that, indicates that the postponed case is not treated differently. It was obviously so, since no one had ever suggested otherwise.
Magen Avraham (OC 554 sk 9) used the leniency for a circumcision to justify ruling leniently, against Maharshal, like the opinions in Rishonim that a postpartum woman has 30 days to recover before becoming obligated to fast on Tisha Bav, instead of 7 like on Yom Kippur. While notably the first basis for any leniency for women on a postponed Tisha Bav, implicit again is that nursing after the formal recovery days is insufficient to warrant categorical leniency from a postponed fast. This is likely what the Shevut Yaakov is expanding when he says he is lenient on a postponed fast for "postpartum women within 30 days and pregnant women who feel somewhat ill".
For some more modern names, R. Yitzchak Ratsabi (Olath Yitschaq 1:67) treats a postponed Tisha Bav like any other. R. Ben-Tzion Abba Shaul (Or Letzion 3:29:3) did as well. R. Moshe Shternbuch (Teshuvot veHanhagot 2:253) fundamentally agrees but feels unable to prevent women from relying on the Beiur Halakha, especially in hot climates (where doctors told him there is significant risk for miscarriage (YMMV)). R. Meir Mazuz is quite hesitant to rely on R. Yosef's leniency. R. Yaakov Ariel allows pregnant women with "great pain" different from what she ordinarily feels on Tisha Bav to break her fast. Of course on the other side there are Poskim who take any leniency they can find with fasting nowadays since we are "weaker" than in previous generations (cf. an extreme example).
Obviously any woman should consult with both her doctor and her rabbi to determine how she should stay safe and follow Halakha, especially regarding any complications with her pregnancy or history of miscarriages or preterm labor. Even if she does need to break her fast, she should, after Havdala on something, eat simple foods and participate as much as possible in the mourning (OC 554:5, 568:12), for those who mourn for Zion will merit to see her rejoicing (Taanit 30b).