Is it legal to own a wombat in the UK?

It is illegal. You cannot own a pet koala anywhere in the world. Koalas are protected, native animals. It does not matter which country of the world you are in: you may not have a pet koala.

In some places, people have koalas living in or regularly wandering through their backyards, but they are not house-friendly animals, and attempts should not be made to interfere with their natural, native activities.

Quite simply, koalas are a protected species. This means it is illegal to have a koala. Few people understand the needs of a koala, and opening up native animals to being owned as pets leads the way to abuse.