This question is deceptively complex. Because random motion is (wait for it) random the molecules move left as much as they move right. The result is that distance the average molecule grows with the square root of time. I'm pulling these number out of my ass, let's say it takes 1 microsecond for the average molecule to move 1 micrometer. How long does 2 micrometers take? Your gut instinct probably says 2 microseconds, but the random back and forth means it doesn't grow linearly.

It would take 4 microseconds to move 2 micrometers (because you doubled the distance and 2 squared is 4). It takes 8 microseconds to move twice as far as that and 16 to move twice as far as that... So to move a millimeter (1000x as far) it takes 1 second (1,000,000 times as long). To move a meter (1e6 micrometers) it takes 1e12 microseconds or 1 million seconds or ~11 days. 2 meters is 44 days, 4 meters is 176 days...

But that's diffusion. You said "stationary body of water" which means diffusion is the only thing acting on it. If only diffusion existed, our sense of smell would take days or even years to detect the fire across the room from us. But if the temperature is different in different parts of the room (like because of a fire or a human), then temperature differences lead to density differences which causes convection. The bulk of mass transport in fluids is caused by convection (wind, currents, etc). You waving your hand in the air is also convection (because your hand creates a pressure difference as it moves)

And just in case my graduated advisor is reading this, there's also migration which is caused by electrical gradients. Migration is relevant in electroplating, batteries, ion channels, and more. But that's like 5 lectures away from what we're talking about now.

Edit: someone in another comment used an actual diffusion coefficient and calculated that it takes years for a molecule to move from one side of a bath tub to the other. It's also worth pointing out that is the mean distance from the source, so half the molecules didn't move that far.