re: O-T weathermen - how good are we at predicting tsunami waves?

I would say in open deep water environments we are really good and can do it really fast (minutes for an entire ocean in a high performance computing environment). U.S. Navy has an ADCIRC model (Google it) for entire oceans for operational forecasting that is very, very accurate worldwide.

The problem comes in when you have to transform waves from deep to shallow water as they shoal and diffract. You have to have more resolution in the models to capture how waves interact with complex coastal geographies and the built environment like seawalls, jetties, etc. to get an accurate depictions of how high water may run up on a coast, but those complexities are sortof whitewashed with very conservative factors of safety (overestimates) anyway to get people out the way.

I wouldnt be surprised if people have thrown millions of pre-baked, pre-run scenarios at AI/ML to speed up the forecasting from a surrogate model in real time.

Long story short, I would say very fast, likely as fast or faster than the wave travel speed in these emergency situations. if there is a real-time forecasting center already set up. The problem becomes more of the speed at which humans can digest the computer outputs and effectively disemninate the information people need to outpace the wave.