Henra, wouldn't all water trapped in the pitot system that was above its temperature flash point flash to steam, and the last bit of water to flash to steam as the pressure increased would be the hottest water that remained in the system after the pressure rise shut off the flashing of the slightly cooler water.
Machinbird,
Yes all wtaer above that temperature (which depends on the ambient pressure) would also evaporate. My point is that the temperature of water and air in more 'remote' locations in the system, e.g. in the tubing would simply be colder and in order to heat it up, that would take some time and would also cool down the steam in the system accordingly.
Yes after a certain time there would be equilinrium, but it's the heat transfer not the pressure transfer that would take time.
It could happen rather quickly however, if the only water in the system was around the heater itself, nothing in the tubing. In that case the steam would have indeed a pressure close to equilibrium pressure for the temperature of the heater.