I started Taskmaster Australia, and noticed they're in the same house as New Zealand!

In which country is the house, who used it first, and how did the second country come to use it?

As far as I remember the house is/was a corporate retreat in NZ. The New Zealand Taskmaster came first, so they 'established' the place.

I guess it just makes sense from a production perspective for the Aussi show to use the same location as TM NZ.

I suppose so. I presume they share a lot of their film industries.