Does it get contaminated with something like the cooling water of a nuclear power plant?

This doesn't happen unless the reactor was sabotaged. Cooling water that interacts with the core is always a closed-loop system. For exactly this reason.

It evaporates. A lot of datacenters use evaporative cooling. They take water from a useable source like a river, and make it into unuseable water vapor.

Search for “water positive” commitment. You will quickly see it's a "goal" thus it is consequently NOT the case. In some places where water is abundant it might not be a problem, where it's scarce then it's literally a choice made between crops to feed people and... compute cycles.

But even then, is the water truly consumed?

Yes. People and crops can't drink steam.

Does it get contaminated with something like the cooling water of a nuclear power plant?

That's not a thing in nuclear plants that are functioning correctly. Water that may be evaporated is kept from contact with fissile material, by design, to prevent regional contamination. Now, Cold War era nuclear jet airplanes were a different matter.

Or does the water just get warm and then either be pumped into a water body somewhere or ideally reused to heat homes?

A minority of datacenters use water in such a way Helsinki is the only one that comes to mind. This would be an excellent way of reducing the environmental impacts but requires investments that corporations are seldom willing to make.

There's loads of problems with the energy consumption of AI, but I don't think the water consumption is such a huge problem? Hopefully, anyway.

Unfortunately, it is. Primarily due to climate change. Water insecurity is an an issue of increasing importance and some companies, like Nestlé (fuck Nestlé) are accelerating it for profit. Of vital importance to human lives is getting ahead of the problem, rather than trying to fix it when it inevitably becomes a disaster and millions are dying from thirst.

In addition to all the other comments, pumping warm water into natural bodies of water can also be bad for the environment.

i know of one nuclear powerplant that does this and it's pretty bad for the coral population there.