It seem realistic that a sedentary population would begin to cultivate Acanthosicyos and a domesticated version would rise. I think spread of the plant are pretty much limited to South Africa, where I could see it spread with the Namib culture, at least until the Europeans arrives and spread it (it could be interesting to see it spreading to Australia).
Of course a lot depend on its robustness, calories per acre, whether it can survive in areas without sea fog and how it compare to other crops. Lots of crops have failed to spread or even fallen out of use, because another crop was better.
Actually the populations OTL were already semi-sedentary and "paracultivation" was proposed to already be a process in action given the two disparate locations Nara horriendus is found in.
This TL starts out with strandloopers turned sealers that also get a taste for whale, dolphin and shark.
They have their own timeline for water craft and expansion of hunting grounds. As time goes on trade amongst themselves, amongst herders and other populations around the coasts intensifies.
Nara develops as a reserve in lean times (given shellfish and scale fish are avoided, more on that later).
The only conditions that's necessary for OTL Nara is cool nights, cool or hot days, accessible subterranean water, good drainage and limited hot humidity.
However ATL Nara is in fact a complex series of crossed and backcrossed mutants of OTL Nara and Gembok Cucumber found further inland, it's the sole crop and has been cultivated so long there is a lot of variation and flexibility such as annual and perennial growth, early and late flowering, frost tolerances, weediness, etc...
Anywhere Canary Islands Date Palm can thrive (not just survive and not hybrids) this ATL nara will grow.