The Ashkenazim Minhag is to say Pirkei Avot all along the summer and Barchi Nafshi after Shabat Mincha in winter. Usually the only gaps are Shabat Tshuva and Shabat Hagadol, when there is used to be the Rabbi speech to all the community. However, sometimes there are gaps like this year. The previous Shabat we said neither Pirkei Avot nor Barchi Nafshi. My question is why the Minhag to say Barchi Nafshi starts only from Bereshit Sidra and on and not from the first Shabat that we stop to say Pirkei Avot (and is not Shabat Tshuva)?
I'm not sure if by "why" you mean "what's the reason behind it" or "what's the source for this custom". If it's the latter, the Rema to Shulchan Aruch Orach Chaim 624:5 writes:
אין אומרים תחינות ולא צו"ץ מי"כ עד סוכות
We don't say Tachanun or Tzidkasecha from Yom Kippur until Sukkos
The Mishnah Berurah (§ 18) adds:
ולא פרקים ולא שיר המעלות
And not Pirkei Avos, nor Shir HaMa'alos (Barchi Nafshi)
So I'm not sure where your premise came from but he says we say neither from Yom Kippur until Sukkos, which would seemingly include Shemini Atzeres and Simchas Torah. Thus, the first week we would say Barchi Nafshi would be Shabbos Bereishis.
Now, looking into the reasons, the Rema in his Darkei Moshe (ad. loc.) cites the Maharil. There, the Maharil writes that we don't say Tachanun between Yom Kippur and Sukkos, because those four days are a gift to the Jews, that their sins aren't counted, as it says in the Midrash (Yalkut Shimoni § 651, quoting Midrash Tanchuma Emor § 22) that Sukkos is the beginning of the accounting of sins. According to this, the Rema only meant the four days between Yom Kippur and Sukkos. This also doesn't seem to explain why not say Barchi Nafshi.
Now, the Mishnah Berurah in his Sha'ar HaTizyun (§ 22) cites what he writes about Pirkei Avos and Barchi Nafshi from Derech HaChaim, in the name of Eliyah Rabbah. I don't have access to Derech HaChaim, but the Eliyah Rabbah (§ 14) says not to say it, as "he" writes at the end of Simman 669. Now, the Eliyah Rabbah is a commentary on the Levush, so I assume he's referring to him. Indeed, I see the Levush at the end of Simman 669 writes extensively about this. I don't know how to provide an accessible link to it (here's Otzar HaChochmah, which isn't free), but I'll summarize what he writes:
He says we start saying Barchi Nafshi on Shabbos Bereishis. The reason is because the entire chapter of Barchi Nafshi discusses creation. We subsequently read it every Shabbos afterwards, in order to remember the act of creation. He says the 15 Shir HaMa'alos are for the same reason.
He then says the reason we don't say Barchi Nafshi or Pirkei Avos on a Shabbos which has Yom Tov on it, is because there's a rule Yom Tov has to be חצי לשם וחצי לכם, half for Hashem and half for you. Since we say Barchi Nafshi after Mincha, that occurs in our half. This is unlike all other Shabboses, as they're entirely dedicated to Hashem. He says that since if Yom Tov would be on Shabbos we wouldn't say them, we also don't say them on Shabbos Chol HaMoed. He says this is also why we don't say them in the Shabboses between Rosh Hashanah, Yom Kippur, and Sukkos. Honestly, I don't get this last point.
For what it's worth, in my Artscroll Yom Kippur Machzor, in the "Laws" section, the very last law (§ 158) says we don't say Tachanun, Av Harachamim, Tzidkasecha, Pirkei Avos, or Barchi Nafshi, between Yom Kippur and Sukkos, as these days are the days that the Temple was dedicated (Moed Katan 9a). They cite the Rema and Mishnah Berurah, although these sources didn't say this reason. It does appear in Biur HaGra, however (he also brings the reason of the Maharil). This reason doesn't seem satisfying, as why not say Pirkei Avos or Barchi Nafshi? Perhaps they only meant the Tachanun etc. which are mournful (this is clearly the intent of the Gra, who doesn't mention Pirkei Avos/Barchi Nafshi). In any event, this also only explains up until Sukkos.