The most accurate water clocks have a design that keeps the height of the water constant. You do this with a second reservoir above the one that water actually drips out of, and then either feed it faster than it drips with an overflow that drains outside the measuring basin, or use a float valve like the one in a toilet tank.
I actually suggested this to my sister when she had to answer the question about building an accurate water clock. We were both in grammar school – it just seemed to be the obvious answer.
But the implied question here is “how does changing gravity affect the water clock?” Even if you do have constant depth in your reservoir.
Keeping the depth constant, would the flow rate not still be proportional to the square root of gravitational acceleration, just as with the hourglass? Just assuming the pressure will be proportional to the acceleration (and the depth, which is not changing).
Probably, though viscosity might complicate things.
That’s a good point, but it’s possible that the limiting factor on earth is the possible flow rate through the hole and not the gravity. So gravity will speed up flow, but only to a point.