By the water analogy, the water pressure at one end of a pipe must be different from the pressure at the other end, for there to be a flow of water through it.
With different pressures at each end, water pressure inside the pipe, at various points, must have some value between those extremes. Pressure doesn't stay high all the way until the other end, and then suddenly drop, it's a gradual decrease of water pressure as you "travel" along it from the high pressure end to low.
Your voltmeter, with one end connected the potentiometer wiper, is simply measuring the the difference in "pressure" between one end (of known, fixed pressure) and some arbitrary point along the pipe corresponding to wiper position.
Current is the same all the way along the resistive path, along the entire length of it, in the same way that if one litre per second enters one end of the pipe, then one litre must exit the other end each second (unless there's a leak, or the pipe is ballooning).
Don't fixate on speed, if you insist on using the water analogy, think in terms of litres per second of flow.
I prefer a road traffic analogy; picture two roadways, a single-lane one, and a three lane one. If I tell you that an observer counts 10 cars passing his position each minute, in both scenarios, then it's easy to see that cars in the single lane road must be travelling 3 times faster than cars in the three-lane road.
In these analogies, current is not speed, current is cars per minute, or litres per second. In electronics, current is charge per second passing some point along a conductor, so the analogy is not a bad one. An ammeter is like the roadside observer, counting cars that pass his position, and reporting how many do that each minute.
Let me extend the road traffic analogy a little. Let's treat voltage as people's motivation to travel. There's a new show in town, everybody want to see it, so they all travel into town, along the three lane highway. Let's change polarity. While they are watching the show, they get news that the town's about to get nuked. Now they all travel, much, much faster along that highway, in the other direction. Same people, different speed and direction.
While speed is obviously a factor, the only good measure of influx or evacuation is how many cars "arrive" per minute, which is only indirectly related to their speed.
And, while speed plays a role in that figure, motivation does too. Assuming no accidents, and an orderly migration of people, cars per minute is a function of both motivation and the road's width.
In this analogy, how does the road know how many cars to pass in a minute? It doesn't. Do the drivers all know how long the journey will take? No. Did the decision to evacuate hinge on any consideration of lane count or distance? No. What happens is simply that people go where they want, at a rate ultimately outside of their control, along paths which they can't change. The resulting relationship between traffic flow, lane count and motivation is determined by all three factors, simultaneously. No individual element in this system has any clue about any of the others, but the system settles into some equilibrium which we call Ohm's law.
