After the announcement of "Mashiv HaRuach U'Morid HaGeshem" before Musaf, even someone who has not yet prayed Shacharit should recite "Mashiv HaRuach U'Morid HaGeshem" in their Shacharit, provided that there is no one praying in another Shacharit minyan.
ב) [1]באמירת מש[ה] ומה"ג[2] גם בשחרית לאחר ששמע ההכרזה (ולפענ"ד רק אם אין מתפלל במנין אחר. ודלא כספק הבה"ל. ואכ"מ) - כ"פ גם בשו"ת שאילת שלום מהד"ת סו"ס קצז ע"פ הירושלמי, ומסיים: וז"פ וברור והבה"ל לא ראה ס' הנ"ל מדלא הזכירם.
This shiur explains that after the announcement of "Mashiv HaRuach U'Morid HaGeshem" before Musaf, even someone who has not yet prayed Shacharit should recite "Mashiv HaRuach U'Morid HaGeshem" in their Shacharit, as long as there is no one praying in another Shacharit minyan. This follows the ruling found in Shut Sheilat Shalom (Mahadurat Devar Torah, end of siman 197) based on the Yerushalmi, and concludes that this is clear and obvious. The Biur Halacha did not see this source, as he does not mention it.
Summary
One who prays Shacharit after the public announcement of "Mashiv HaRuach U'Morid HaGeshem" should include it in their prayer, provided no other Shacharit minyan is still praying.