Islamic Ruling on Delaying Sajdah Tilawah After Reciting an Ayah of Sajdah
In Shafi‘i fiqh, Sajdah Tilawah is Sunnah immediately after reciting the Ayah of Sajdah. If a long gap occurs, it is no longer Sunnah.

According to Shafi‘i fiqh, it is Sunnah to perform Sajdah Tilawah right after reciting the Ayah of Sajdah.
If a long gap happens between the recitation and the sajdah, then Sajdah Tilawah is no longer Sunnah. This is because its proper time has passed.
Imam al Khatib al Shirbini رحمه الله states in Mughni al Muhtaj that if a person does not perform sajdah immediately after finishing the Ayah of Sajdah, and a long delay occurs even due to an excuse, then he should not perform it as adaa. There is also no qada for it because it is connected to a specific cause, like eclipse prayer. However, if the delay is short, then Sajdah Tilawah is still Sunnah.
This fatwa clearly explains the Islamic ruling that Sajdah Tilawah must follow the recitation closely. If the gap is short, it remains Sunnah. If the gap is long, it is no longer prescribed.