How much time is there between an underwater earthquake and the moment a tsunami hits the coast?
As a general guideline, the time between the actual earthquake and the time the tsunami arrives is on the order of minutes to hours.
Tsunamis out in the deep ocean travel very fast -- around 500 mph -- which means that they cross ocean basins in about as much time as it takes a plane to fly across. So an earthquake in Japan gives Hawaii and the US West Coast many many hours of warning.
Close to shore, tsunamis are much slower, but of course the distances are also shorter. For a place where earthquakes happen close to shore because of a subduction zone, say in Japan, this only gives you a few or a few tens or minutes of advance warning. If you take the 2011 tsunami in Japan, warnings were sounded quite quickly, and that allowed a large number of people to get to higher ground and will have saved tens of thousands of lives. But at the same time, it was not enough time to actually go door to door, put people onto buses, and get them out of harm's way if either they did not hear the warning, or refused for whatever reason to immediately evacuate.