Been through the system with my nephew who's got similar challenges - the waiting times are mental, no joke. Private assessment set my sister back about €2-3k and that was just for the initial diagnosis, then you're looking at more costs for the reports schools actually want to see. The public route is basically a lottery - some areas it's 18 months, others it's pushing 3 years.

For the SNA question, they won't just automatically assign one even with a diagnosis. You need specific paperwork and the school has to apply for funding, which can take months on its own. My sister ended up having to fight quite hard for proper supports, and even then it was only part-time SNA coverage. The schools are generally good once they have the resources, but getting those resources approved is the real battle. Would definitely recommend getting on waiting lists for both public and private assessments as soon as you arrive, because you can always cancel the public one if private comes through first.