An underworld could have climate zones... In a way, you've already partly defined it with measuring that the higher zones are warmer then the colder zones.

The next question would really be, how much water does your world have? Does the water heat up closer to the core? As the water heats up, do the natural warm air flow cause it to flow down tunnels to lower levels? Is there a cold ridge in a cavern that causes the warm air to condense and rain? A simplified version of this can get a good deal of weather up to the coldest areas.

I don't think you'd have snow, but ice and sleet would probably be common. the further up you get.