This doesn't have to be one federalized nation but at least some sort of formal union exclusively between these two nations.
The only way I could see this happening is a global revolt against the British Empire. Such a civil war or a (the more likely) revolution occurring would have to take place before 1945
Germany sets up a puppet government in London.
There is also literally no reason why either would want to pursue this when Australia can Federalize with New Zealand and British Oceania possessions, whilst Canada can do the same with the Caribbean and Bermuda.