It's quite plausible to suggest that Moshe's birth which lit up the room was known to more than just the family. Also, the fact that Yocheved weaned him for 2 years then sent him to the palace could have been quite exciting for the Jewish People hoping that he would rise to greatness. Obviously when Moshe returned from Midian he had to provide signs to prove that he was the redeemer, but was there ever any excitement or expectation at any time that Moshe as he was growing up or beyond that he was going to be the redeemer? What I meant to ask is, when Moshe came back on the scene did the people of Goshen say "I always knew it!"
The Gemara says that Miriam prophesied to her father Amram that Moshe would save the Jews, but when he was put in the river Amram gave up hope.
מרים דכתיב (שמות טו, כ) ותקח מרים הנביאה אחות אהרן ולא אחות משה אמר ר"נ אמר רב שהיתה מתנבאה כשהיא אחות אהרן ואומרת עתידה אמי שתלד בן שיושיע את ישראל ובשעה שנולד נתמלא כל הבית כולו אורה עמד אביה ונשקה על ראשה אמר לה בתי נתקיימה נבואתיך
Miriam was a prophetess, as it is written explicitly: “And Miriam the prophetess, the sister of Aaron, took a timbrel in her hand” (Exodus 15:20). The Gemara asks: Was she the sister only of Aaron, and not the sister of Moses? Why does the verse mention only one of her brothers? Rav Naḥman said that Rav said: For she prophesied when she was the sister of Aaron, i.e., she prophesied since her youth, even before Moses was born, and she would say: My mother is destined to bear a son who will deliver the Jewish people to salvation. And at the time when Moses was born the entire house was filled with light, and her father stood and kissed her on the head, and said to her: My daughter, your prophecy has been fulfilled.
וכיון שהשליכוהו ליאור עמד אביה וטפחה על ראשה ואמר לה בתי היכן נבואתיך היינו דכתיב (שמות ב, ד) ותתצב אחותו מרחוק לדעה לדעת מה יהא בסוף נבואתה
But once Moses was cast into the river, her father arose and rapped her on the head, saying to her: My daughter, where is your prophecy now, as it looked as though the young Moses would soon meet his end. This is the meaning of that which is written with regard to Miriam’s watching Moses in the river: “And his sister stood at a distance to know what would be done to him” (Exodus 2:4), i.e., to know what would be with the end of her prophecy, as she had prophesied that her brother was destined to be the savior of the Jewish people.
However according to the fuller account of the story in the Mekhilta Miriam did not give up hope.
תקח מרים הנביאה, וכי היכן מצינו שנביאה היתה מרים אלא שאמרה לאביה סופך אתה מוליד בן שמושיע את ישראל מיד מצרים מיד וילך איש מבית לוי ויקח וגו' (שמות ב) ותהר ותלד בן וגו' ולא יכלה עוד הצפינו. נזף בה אביה אמ' לה בתי היכן נבואותיך ועודה מחזקת בנבואתה שנ' ותתצב אחותו מרחוק לדעה מה יעשה לו ואין יציבה אלא נבואה שנ' ראיתי את ה' נצב על המזבח (עמוס ט) ואומר ויבא ה' ויתיצב (שמ"א ג) ואומר קרא את יהושע והתיצבו וגו' וילך משה ויהושע ויתיצבו באהל מועד (דברים לא): מרחוק אין מרחוק בכל מקום אלא רוח הקדש שנאמר מרחוק ה' נראה לי (ירמיה לא): לדעה אין דעה אלא רוח הקדש שנ' כי מלאה הארץ דעה את ה' וגו' (ישעיה יא) [ואומר כי תמלא הארץ לדעת את כבוד ה' כמים יכסו על ים (חבקוק ב)]: מה יעשה לו אין עשייה אלא רוח הקדש שנ' כי לא יעשה ה' אלהים דבר וגו' (עמוס ג):
(Ibid. 20) "Then Miriam the prophetess took": Where do we find that Miriam was a prophetess? She said to her father (Amram): In the end, you will beget a son who will be the savior of Israel from Egypt — (Ibid. 2:1) "And there went a man (Amram) from the house of Levi and took the daughter of Levi, (Yocheved — after he had separated from her because of Pharaoh's decree to drown the males) (2) "And she conceived and bore a son … (3) and could conceal him no longer…" — whereupon her father rebuked her, saying: My daughter, where is your prophecy? But she stood fast to her prophecy, viz. (4) "And his sister stood from afar to know what would be done with him," "standing" ("yetzivah") signifying prophecy, viz. (Amos 9:1) "I saw the L rd standing on the altar", and (I Samuel 3:10) "And the L rd came and He stood, etc.", and (Devarim 31:19) "Call Joshua and stand, etc." (Exodus, Ibid.) "from afar": "from afar" everywhere signifies the Holy Spirit, viz. (Jeremiah 31:2) "From afar the L rd appeared to me." (Exodus, Ibid.) "to know": "Knowing" signifies the Holy Spirit, viz. (Isaiah 11:9) "for the earth will be filled with knowing of the L rd, etc." (Exodus, Ibid.) "what would be done with him": "Doing signifies the Holy Spirit, viz. (Amos 3:7) "For the L rd G d will not do a thing unless He has revealed His secret to His servants the prophets." (Exodus 15:20)
And since Miriam reunited Moshe with his mother (here), perhaps they still knew he would be the savior of her prophecy.
However, Moses himself seemed to be unaware of this prophecy:
And Moses said to G-d: Who am I that I should go to Pharaoh [i.e., How am I worthy that I should speak with kings], and [how have the Jews merited it] that I should take the children of Israel out of Egypt? Exodus 3:11
And I would argue that the necessity of Moses providing signs implies that the prophecy of Miriam was not widely known at all.
Surely not. The Torah tells clearly that Moses knew this fact himself:
וַיַּ֤עַן מֹשֶׁה֙ וַיֹּ֔אמֶר וְהֵן֙ לֹֽא־יַאֲמִ֣ינוּ לִ֔י וְלֹ֥א יִשְׁמְע֖וּ בְּקֹלִ֑י כִּ֣י יֹֽאמְר֔וּ לֹֽא־נִרְאָ֥ה אֵלֶ֖יךָ ה׃
But Moses spoke up and said, “What if they do not believe me and do not listen to me, but say: The LORD did not appear to you?” Exodus.4.1-2
Moreover, when he came to the people, it was Aharon who told the story and whom the people believed:
וַיְדַבֵּר אַהֲרֹן אֵת כָּל־הַדְּבָרִים אֲשֶׁר־דִּבֶּר יְהוָה אֶל־מֹשֶׁה וַיַּעַשׂ הָאֹתֹת לְעֵינֵי הָעָם׃ וַיַּאֲמֵן הָעָם וַיִּשְׁמְעוּ כִּי־פָקַד יְהוָה אֶת־בְּנֵי יִשְׂרָאֵל וְכִי רָאָה אֶת־עָנְיָם וַיִּקְּדוּ וַיִּשְׁתַּחֲוּוּ׃
Aaron repeated all the words that the LORD had spoken to Moses, and he performed the signs in the sight of the people, and the people were convinced. Exodus.4.30