Short answer
No. Aisha never had children. She took the kunya (honorific name) Umm Abdullah after her nephew Abdullah ibn al-Zubayr, with the Prophet's ﷺ permission. Khadijah was the only one of his wives to bear him children; his son Ibrahim was born to Mariyah.
The Sunni historical account is that Aisha had no children. All of the Prophet's ﷺ children were from Khadijah, except Ibrahim, whose mother was Mariyah al-Qibtiyya.
The four schools agree on this.
Explanation
Aisha bint Abi Bakr did not give birth at any point in her life. So a question like "how old was Aisha when she gave birth" has no answer: it did not happen.
In Arab culture a woman is honoured by being called "mother of" her eldest son. Aisha wished for such a name. She asked the Prophet ﷺ about it, and he told her to take the name of her nephew, Abdullah ibn al-Zubayr, the son of her sister Asma. From then on she was called Umm Abdullah. Abdullah was the first child born to the Emigrants (Muhajirun) after the Hijrah.
Her motherhood took another form. Allah calls the Prophet's ﷺ wives the mothers of the believers. Aisha raised and taught orphans and relatives, and she taught the ummah. Her students included Urwa ibn al-Zubayr, al-Qasim ibn Muhammad and many others who became pillars of knowledge in Madinah.
The Prophet's ﷺ children were Qasim, Zaynab, Ruqayyah, Umm Kulthum, Fatimah and Abdullah from Khadijah, and Ibrahim from Mariyah. See How many children did Prophet Muhammad have? and the family tree.
The evidence
Quran, Surah Al-Ahzaab 33:6
ٱلنَّبِىُّ أَوْلَىٰ بِٱلْمُؤْمِنِينَ مِنْ أَنفُسِهِمْ ۖ وَأَزْوَٰجُهُۥٓ أُمَّهَٰتُهُمْ ۗ وَأُو۟لُوا۟ ٱلْأَرْحَامِ بَعْضُهُمْ أَوْلَىٰ بِبَعْضٍۢ فِى كِتَٰبِ ٱللَّهِ مِنَ ٱلْمُؤْمِنِينَ وَٱلْمُهَٰجِرِينَ إِلَّآ أَن تَفْعَلُوٓا۟ إِلَىٰٓ أَوْلِيَآئِكُم مَّعْرُوفًۭا ۚ كَانَ ذَٰلِكَ فِى ٱلْكِتَٰبِ مَسْطُورًۭا ﴿٦﴾
The Prophet is more worthy of the believers than themselves, and his wives are [in the position of] their mothers. And those of [blood] relationship are more entitled [to inheritance] in the decree of Allah than the [other] believers and the emigrants, except that you may do to your close associates a kindness [through bequest]. That was in the Book inscribed.
Aisha is a mother of all believers, though she had no children.
Sunan Abi Dawud 4970
Grading to be checked... فاكتني بابنك عبد الله ...
"Take your kunya from your son Abdullah."
He told her to be called after her nephew, whom he called her son.
Where the ruling is recorded
- al-Isaba fi Tamyiz al-Sahaba (Ibn Hajar), Entry on Aisha bint Abi Bakr. Notes that she had no children and took the kunya Umm Abdullah.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the first year after the Hijrah, birth of Abdullah ibn al-Zubayr. Records him as the first child born to the emigrants in Madinah.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.