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.