May you please inform me about the permissibility of Eid takbirat especially in the mosques and after the obligatory prayers? May you also please put emphasis on the hadith reported by Umm Atiyyah. And may Allah reward you. تكبيرات العيد

Praise be to Allah, and may Allah’s peace and blessings be upon the Messenger of Allah, his family and his companions.
As to what follows:
With regards to answering your question we say, success is by Allah Almighty.
Saying takbir aloud after swalah on Eid is a sunnah that is narrated by the companions, but the loudness should not get to the level of harming and disturbing people. It should be done in a way that will let the Muslims know, especially these days which are days of making dhikr.
The hadith of Umm Atiyyah you mentioned is found in Muslim on the authority of Asim from Hafsa from Umm Atiyyah who said that:
“He (the Messenger of Allah) commanded us that we should take out unmarried women and purdah-observing ladies for Eid prayers, and he commanded the menstruating women to remain away from the place of worship of the Muslims.”
Another narration says:
“We were commanded to bring out on Eid days purdah-observing ladies and those unmarried and menstruating women came out but remained behind people and pronounced takbir (Allahu Akbar) along with them.”
This hadith is the proof and it is in Muslim. Furthermore it indicates that saying takbir in the masjids and places of prayer was the slogan of that day. The permissibility of raising the voice when saying the takbir is indicated by Umar’s (may Allah be pleased with him) action. Bukhari narrates that Umar (may Allah be pleased with him) used to say takbir in Mina, and the people in the masjid said takbir , as well as the people in the markets until mina vibrated with takbir.