Al-Jallad talks a bit about Nabatean Aramaic here: https://15minutehistory.org/2016/04/27/episode-82-what-writing-can-tell-us-about-the-arabs-before-islam/
So the Nabataeans are conquered by the Romans in 106 CE. Throughout this period they are writing primarily in Aramaic, even though they are speaking Arabic. After the fall of their kingdom, the Nabataean script and writing tradition, the use of Nabataean Aramaic, the particular kind of Aramaric survives in northern Arabia, and you start to see the intrusion of more and more Arabic. In 328 CE, an epitaph was produced for a king named Mal Qays or Imrou Qays, we can’t exactly determine the pronunciation, son of ‘Amrou who calls himself “Malek al arab kulla“—King of all the Arabs, and this is in the Nabataean script, but the language, which, except for the word for son—they use the aramaic bar—is Arabic, completely Arabic. So we see, at least at this point, the use of the Nabataean script to write Arabic language, and in a context of prestige. And, basically between the 3rd century CE and the 5th century CE, a great scholar of Nabataean epigraphy, Layla Nehmé identified this as a transitional period between the Nabataean Aramaic script and what is recognizably the Arabic script. And the language of this period is mostly Aramaic, although there are more and more Arabic intrusions. So this would be the beginning of the Arabic script as we know it. Now, why this script then spread and replaced the indigenous alphabet of Arabia that was used for over 1000 years is unclear.