The bucket will not be able to scoop up Lava until it is upgraded once, when it reaches Bucket Strength 2. Likewise, it will not be able to scoop up Radiation until it reaches Bucket Strength 3, after being upgraded to Level 3.
If you switch your ability and put the bucket away, it will retain its liquid.
Using the skill on any other biome with a full bucket will result in the message "LIQUID DUMPED" and will empty your bucket, wasting the skill.
The correct way to use this is to use the bucket in any of the 3 biomes in which the bucket can be used on, go to one of the other two and dump the liquid (Example: Using the bucket in the Ocean and then dropping the water in the Volcano will result in Obsidian).