Since the speed of the water may vary across the river, let us focus on the speed at the river mouth.
When a water wheel is placed in a river, part of the kinetic energy of the water is stored. Therefore, the water should flow slower than without the water wheel. This is the case for every point between the water wheel and the river mouth.
On the other hand, the total amount of water that enters the river annually (through glaciers, rain, etc) remains the same. Let us call this amount A. Now the total amount of water that exits the river anually is also A. Let $\sigma$ be the area of the cross-section of the river at the mouth. Now the speed of the water at the cross section is proportional to $A/\sigma$. This is independent on the water wheel. Therefore, the river does not flow slower with the water wheel.
How can these two be reconciled?
UPDATE I would like to emphasize that my question is NOT about the flow rate of the water, but about the velocity. Here flow rate is in liters per second, while velocity is in meters per second.
Also, the question is about the water downstream of the water wheel, not upstream.
Finally, I would like to know what the difference is between the situation without the water wheel, and the situation with the water wheel when equilibrium is attained.