At least in my field people work with AI to write code and find information, so I assume whoever is making the data analysis for James Webb has done the same.

For using neural networks to actually analyse the data I’m not sure, because the goal of these analyses is to understand, and not just to classify. That being said, for known phenomena with lots of training data, neural networks may help. But (not really knowing, but based on my hunches) I would say that they use some explainable and understandable stochastic methods to find candidates for something that was never observed before, such as a black hole star.

Feel free to do a literature search and prove me wrong though :)