To what extent is the ḥadīth sound, narrated by Lady ‘A’isha (may Allah be pleased with her) from the Messenger of Allah (peace and blessings be upon him), regarding the location of people on the Day the earth is changed to other than the earth and the heavens, and that they will be on the Ṣirāṭ, along with a mention of the Ṣirāṭ's characteristics and what will occur on it?

The content mentioned in the question is not a prophetic hadith verbatim. Rather, it is an explanation of the Sirat (the bridge over Hell) that combines several hadiths from the Prophet, peace and blessings be upon him. Among these are what Abu Sa'id Al-Khudri narrated concerning a bridge, "slippery and shaky," over Hell, upon which are hooks, grappling irons, and Sa'dan thorns. Believers will pass over it at varying speeds, with some being saved, some scratched, and some piled into the Fire. Also, what Abu Hurayrah narrated about the Sirat being placed over the middle of Hell, and the Prophet being the first to cross it with his nation, and that it will have hooks that snatch people according to their deeds. And that Amanah (trustworthiness) and Rahm (kinship) will stand on its two sides. The Sirat is finer than a hair and sharper than a sword, and this is established in meaning and content, even if not with all the exact wording.