If your house is a perfect cylinder and the ventilator is not to strong, there may be a chance to observe a pressure wave propagating from one windows to the other and then a current of air establishing itself.
This scenario is however not realised in reality because the flow of air becomes turbulent as a result of the complicated geometry of houses. Then the main effect of the ventilator is to mix the air in the region where it is active. The overall flow is rapidly damped and very probably does not reach the other window. It might be possible to observe a turbulent front propagating from one room to the other. This may lead to a slow change of the temperature as the air inside the house gets mixed with the outside air.
You will however have to work very hard if you really want to answer your question quantitatively. You could set up a numerical simulation of Navier--Stokes equations with the geometry of your home (and the fan) accounted for, but I would advise against it. Turbulence is an extremely complex phenomena that is difficult to simulate and more or less precludes analytical solutions.
The easiest way to answer your questions would be do just do it: Set your ventilator on a timer, go to the other window with a stopwatch and measure how long it takes for the air to feel different.