A watt does have time! It's equal to 1 Joule per second of energy transfer. That's also why power consumption is often measured in Kilowatt-Hours, rather than just Kilowatts, by electric meters. 1KWh is equal to 1000 watts of power being used for 1 hour.

Using the common water analogy you mentioned, the way I like to think of it is that Volts are like the water pressure, amps are the speed (or flow rate) of the water, and watts are the total water moved per second.

In reality, the only difference is that instead of moving water, we're moving electrons. That's why the water analogy is so effective.

I suppose this doesn't entirely answer your question, since I'm still using the water analogy, but I genuinely can't think of another way to explain it. I'll ponder for a while, maybe I can think of another analogy with some time.

the Watt is never written with respect to time, why is that?

Power is energy over time and, like @derekiscool said, 1 W = 1 J/s, so it’s just inherently there. Horsepower is also a measure of power, equal to 745.7 W or 745.7 J/s, but you probably never question why cars aren’t rated in “hp/second” or something like that. It’s the same idea.

I'm going to try a different analogy, maybe that helps you a bit!

You asked about 3 things, watts (power), amps (electric current) and volts (electric potential).

Current (amps)

I'll start with current, since that's the easiest. I'm going to operate under the assumption that you know what electrons are. Current is how many electrons flow / go in a particular direction. Amps is the international unit we use today, but it could be anything, really. It's some factor of "number of electrons / second" going in this direction.

Voltage (volts)

Imagine having a ball on a flat surface. If you just leave the ball there, nothing happens. That's 0 volts, or 0 electric potential. Now if you imagine a hill, if you put a ball anywhere on the slope, that ball will want to roll downhill. By doing some work yourself, you can roll the ball uphill, but it will want to roll downhill normally, and you need to work against that. The steeper the hill, the higher the voltage. You can imagine electric potential like that too, just for electrons and wires instead of a ball and a hill. The wall outlet is generally 230V (or 110V if you're in the US), which is just a metric for "how strong the electrons want to come out of there". This isn't perfectly accurate, but I hope it gets the idea across.

Power (watts)

This is pretty simple too. It's how much work the given thing is doing per second. A watt is defined as "Joule / second", but it's essentially just "energy / time". A fun little addendum, power is voltage * current, so: watt = volts * amps. :)

Watts tell you energy consumption. For example, a light bulb consumes a steady amount of electricity when it’s turned on. A 100 watt light bulb consumes twice as much as a 50 watt bulb, or the same as two 50 watt bulbs.

(When connected in parallel, and ignoring things like dimmer switches.)

As others have said, it’s implicitly per second. You need to multiply by the time that the light bulb is left on to get energy used.

If you want a water analogy, a bigger pipe will let more water flow through it than a smaller pipe. But you could have two smaller pipes that are equivalent to a bigger one.

I'll take a stab at something really short and maybe a bit more conceptual*:
Watts: How much energy is flowing every second
Volts: How strongly motivated electrons are to move
Amps: How many electrons are flowing every second

And then some slightly longer explanations maybe:
Watts is power transfer which comes from the combination of voltage (motivation for electrons to move) and amps (how many electrons are flowing). Watts can also be used for heat flow (how much thermal energy moves every second) which may or may not be helpful for remembering.

Voltage is often called a potential. You could almost think of it like a rubber band or spring. Without pulling it has zero potential, but as you stretch it, there's increasingly greater potential energy that can be released when you let it go for it to jump back to its unexpanded state. Voltage relates how motivated electrons are move from a high voltage ("stretched out") to a low voltage (a resting state).

Amps are measuring the individual electrons as they race from a high voltage to a lower voltage (which is electric current). This is the water in the water analogy. But could also be marbles rolling down a hill. It's counting how many run by every second on their way down.

You didn't mention it, but thinking of resistance also helps intuition. For the rubber band/spring anology, resistance is kind of like the air the rubber band or spring is sitting in. When the spring gets released, air doesn't really resist it snapping back to its original shape. But if you held the spring in honey or molasses and stretched it, you'd see it contract more slowly because of it needing to move really thick liquid out of the way. Resistance slows down the rate that electrons move from high voltage to low voltage. So for the same voltage, it affects how quickly electrons flow and as a result, how much energy moves every second.

*Trying to keep this pretty abstract, I know these aren't especially rigorous explanations.

This might help - https://i.imgur.com/yNMWeDv.jpeg

Watt is the most confusing for me shouldn't it with some unit of time

It is. Watts are just j/s. It's just given a name since it's a common unit. This isn't even a particularly electrical unit, it's used everywhere.

Volts: imagine if you invented a unit, say, P, and say it's equal to g * h, that is, the gravitational acceleration at a specific field distance * the separation between the two objects. If you were to multiply this P by mass, you'd have energy. E = g * m * h. Similarly, if you multiply volts by coloumbs - that is, charge - you get energy. That's why it's usually called electric potential. IF there was charge, how much energy would there be? Or, per every unit of charge, how much energy is in the electromagnetic field?

Amps: charge / second. That shouldn't be too hard to intuit. Maybe more useful, think of what is missing. Amps describes how much charge is going in; does that tell you how much energy is in the system? No, because you don't know how much energy each charge has. Charge is like mass in gravity. You don't know how much energy is in a gravitational system when I tell you just the masses. A car floating 100m in the sky has more energy than a car on the ground - that is evident by the fact that car will fall down and explode if it is 100m in the sky. The height and gravitational constant is required to tell you how much each unit of mass has how much energy.

What tells you how much energy each charge has? Voltage.

I refreshed before posting so I’ll only leave this part:

A watt is 1 J/s so it “is“ per second.

It might help to look at the SI base units and cancel out various units like time (s) to see how V, A, and W relate.

If I may take the wider perspective, I wonder how long it will be until AI completely supplants this kind of post? I’ve been using Claude to cram quiz questions and explain difficult concepts and it’s been going pretty well I think. It’s one of those things that is easy to forget that everyone isn’t doing after a while.

May I ask if there's a context in which this came up recently? Perhaps providing an answer in terms of something that you run into frequently would stick in your head better than something to do with water?