You need to be careful with analogies. Here are some problems in the analogy you describe:
water starts flowing from a collector (battery)
Nothing in an electric circuit really works like a collector of water. In your analogy, water is electric charge which, in metals, is carried by electrons slowly drifting. Batteries do not store charge, they are not a reservoir of charge (nor of electrons).
Batteries store energy in chemical form. A better analogy is that a primary battery is a coal-fired water pump that will deplete it's store of coal as it pumps water. A secondary battery is a bit like a pump powered by a wind-up spring, it can be run in reverse to wind up the spring. These pumps can only pump water if their outlets are connected to a circuit of pipes that eventually returns to their inlets.
does that mean that the voltage and current increase ...
Voltage isn't something you measure at one point, it's something you measure between two points - it's a difference.
If you measure the voltage at every millimeter of the circuit with respect to the batteries negative terminal you will see the voltage monotonically decreasing as you progress around the circuit‡.
The current measured at any point in the circuit‡ is the same. It neither increases nor decreases
... but the number of electrons would decrease
It isn't very useful to think of the number of electrons increasing or decreasing. Where would they go? Where would they appear from?
You measure a current† of water in litres per second. You measure a current of electricity in coulombs per second (amperes). In a steady-state system, this current is the same in all parts of a simple serial circuit - whether of pipes or of water. A constriction in a pipe cannot make n litres per second of water disappear.
If you slightly turn a gate-valve in a water pipe, the flow of water (litres per second) decreases in all parts of the circuit, including in the pump.
resistance increases causing the pressure and speed of the water to increase but the volume would decrease.
That's not how water works!
If we imagine a simple circuit where a water pump is pumping water around a loop of pipe. The pipe is of uniform size apart from one place where we have a section of narrower pipe.
resistance
The resistance is greater in the narrower pipe (a greater proportion of the water is close to the pipe walls and experiencing friction)
pressure
However the pressure is lower, not higher!
speed
It's the lower pressure that causes the water to accellerate to a higher velocity as it enters the narrow section.
volume
When you say the volume increases, I think you mean the velocity increases. Water is relatively incompressible, it's volume doesn't change much at the pressures applying in our analogy.
The flow rate (volume per second) is unchanged.
Footnotes
† This is one of the areas where the analogy starts to break down. The word "current" is used inconsistently. If you asked someone to measure the current in a river they might give you an answer in metres per second ("current" = average drift velocity of H2O molecules) instead of litres per second ("flow" = litres per second passing a fixed point).
‡ This answer applies only to a simple circuit of battery and resistor connected by copper wires.
Best Answer
You seem to have voltage and current conflated.
Voltage is more properly called electromotive force. It does not, in itself, flow, or transfer energy.
Current (usually measured in amperes) is a measure of how much electric charge is moving per unit of time. Current is also not, in itself, a flow of energy.
The flow of energy is called power. To have power, you need both current (\$I\$) and voltage (\$E\$). The power is equal to the product of the two:
$$ P = IE $$
It helps to think about this in terms of analogous mechanical systems, since we can observe mechanical systems directly with our senses. Mechanical systems also have power, where it is equal to the product of force and velocity:
$$ P = Fv $$
If you have force but no velocity, you have no power. An example would be a rubber band stretched between two stationary supports. The band is exerting a force on the supports. This tension is potential energy. But, nothing is moving, and none of that energy stored in the stretched band is being transfered to anything else.
However, if the band can move the supports, now we have velocity. As the band moves the supports, the energy stored in the stretched band will be converted to kinetic energy in the supports. The rate at which this energy transfer happens is power.
Voltage is a force that moves electric charge. Current is the velocity of electric charge. Resistance is how easy it is to move the supports.
Here's a mechanical system that's more analogous to your circuit:
We have a rigid ring, attached to a motor that applies some force to turn it. Also attached to the ring, we have a brake, which resists the turning of the ring. For this analogy to be proper, this has to be a brake that provides a force proportional to the velocity of the ring moving through it. Imagine it's coupled to a fan, so as the ring turns faster, the fan turns faster, creating more aerodynamic drag.
If the motor is applying a force of \$1kN\$, then the brake must be applying an equal force in the opposite direction. If the brake's force is not equal to the motor's, then the ring will experience a net force that will accelerate or decelerate it until the brake's force is equal, and the ring turns at a constant speed. Thus, if the force of the motor is constant, the speed of the ring is a function of the strength of the brake. This is analogous to Ohm's law.
What other forces are acting on the ring? Since we are considering an idealized system with no friction, there are none. If you were to insert strain gauges at points A and B, you would measure a difference between them. B is being compressed as the motor shoves the ring into the brake against its resistance, and A is being stretched as the motor sucks it out of the brake.
But what's the difference between B and C? there is none. If that's not intuitively obvious, consider that you must cut a gap in the ring and insert your hand so this machine can smash it. Is there a point at which you'd prefer to do this? No, your hand will be equally smashed regardless of where you do it on the left side of the ring.
The forces measured by the strain gauges are analogous to voltage. We can only measure voltages relative to some other voltage. That's why your voltmeter has two probes. Wherever you put the black lead is defined as "0V". So, the scenario you present in your question is like measuring the difference between B and C: it is zero.
This seems a little weird, because we know there is a compressive force on that entire side of the ring. It seems like that should be good for something. But consider this: the weight of all the gas in Earth's atmosphere results in a pressure at sea level of about 15 pounds per square inch. Does this mean we can make a machine that's powered just because it's exposed to this pressure? No. In order to do work with this atmospheric pressure, we need a difference in pressure. Without a difference, we can't make the air move. Consider again the definitions of power above, and it should become clear how this is true.