I have cured 3 zombie villagers, 2 I had become farmers and one is a green coat. I had them on my mooshroom island, but they didn't do anything. They wouldn't farm, pick up items or breed. Even after moving them to an (lukewarm) ocean biome, it still does not work. I can trade with them, they do change professions.

This has to be a bug, as there is no interaction mentioned on any wiki based around villagers not doing certain things in certain biomes.