The start time for the Takbeerat for Eid ul-Fitr is when going out for the prayer until the Imam emerges.
Ibn Rushd rahimahullah said:
“They differed (regarding) what time the Takbir for Eid ul-Fitr starts after agreeing that it is recommended due to the statement of the Exalted:
وَلِتُكْمِلُواْ الْعِدَّةَ وَلِتُكَبِّرُواْ اللَّهَ عَلَى مَا هَدَاكُمْ
(He wants that you) must complete the same number (of days), and that you must magnify Allah [i.e. to say Takbir (Allahu-Akbar; Allah is the Most Great) on seeing the crescent of the months of Ramadan and Shawwal] for having guided you
So the majority of the Scholars made Takbir when they go out for the prayer and it is the school of thought of Ibn Umar and a group of Companions and Tabi’een. Also, it was said by Imam Malik, Ahmad, Ishaq, and Abu Thawr. Some people said: he makes takbir in the night of fitr when the moon is sighted until he comes to the musallah and until the Imam comes out”
So there is no need to make the Takbeerat after the prayers for Salahtul Eid because the timing starts when going out to Eid. Whereas for Eid al-Adha it can be done in general after the prayers, before the prayers, and throughout the day. That does not mean in unison though [see this post regarding making takbir in unison]
And Allah Knows Best
Faisal Ibn Abdul Qaadir Ibn Hassan
Shaykh Muqbil rahimahullah was asked when does the takbir of Eid begin:
في عيد الفطر عند خروجهم للصلاة ، وقبل الصلاة ، وأما بعدها فلا بأس أن يكبر يوم العيد نفسه .
“… In Eid al-Fitr when they [i.e. people] are departing for Salat and before the prayer. As for after it, then there is no harm in performing the takbir on the actual day of Eid… ”
Likewise, the advice of Shaykh Albany and al-Uthaymeen that the takbir shouldn’t be confined to right after the five daily prayers and if a person chooses to establish the takbir after it, then he should perform the takbir after completing his prescribed adhkaar. And Allah knows best.
AbdulFattaah Bin Uthman