I'll incidentally mention that gluten-free diets tend to incidentally be lower in FODMAPs which are another gut intervention, and autism has LONG correlated to digestive and dietary issues. Ketogenic/low-carb is also popular.
The gut-brain link I'm aware of three major theories. Gut issues could be causing a disorder of the brain. A disorder of the brain could be causing disordered lifestyle habits (Diet, exercise) causing digestive issues. A disorder of the brain could directly impacting digestive function.
I will however say, I would treat claims about the gut-brain axis and autism with the utmost scepticism as disgraced doctor Andrew Wakefield, who popularised the link between the MMR vaccine and autism to personally financially benefit from an undisclosed patent on a measles vaccine using fraudulent research, was a gastroenterologist and later anti-vax influencer who ended up popularising the gut-brain link of autism particularly among those who distrust scientific institutions. He notably influenced Jenny McCarthy who became a noted alt-medicine momfluencer in the '00s/'10s pushing gluten-free and anti-vax. Which is to say, research on the gut-brain link is done quite a lot by legitimate scientific researchers, quacks, and frauds alike so you have to be discriminating.
Additionally there have been various scientific reviews of dietary interventions, gluten-free being one of the most researched, and the evidence hasn't been shown ANY dietary intervention or gut intervention like fecal transplants is a silver bullet.
This isn't to say that there isn't anything to the idea that some people we've diagnosed as autistic or believe they're autistic, cannot be helped by dietary interventions. I would dare speculate, that a significant portion of those called "autistic" could make autistic symptoms fall into the sub-clinical range through gut interventions (diet AND exercise), and that such people may even be clinically distinct from other people considered autistic other than through their similar behaviour.