Pretty quickly according to figure 1. Britain, Australia and New Zealand quickly had similar standards of living in the middle of the 19th century until about 1880. Australia was richer for about 20 years until the great Federation drought took the wind out of its sails.
Figure 1: British, Australian and New Zealand GDP per capita (1990 Int. GK$PPP), 1820 – 2010
Source: The Maddison-Project, http://www.ggdc.net/maddison/maddison-project/home.htm 2013 version.
New Zealand then broke away at the end of the Second World War from both Australia and UK. In the mid-1960s circumstances changed with Australia drifting ahead of the UK and New Zealand drifting away to a lower standard of living.
Figure 2: British and Australian GDP per capita (1990 Int. GK$PPP), 1820 – 2010
Source: The Maddison-Project, http://www.ggdc.net/maddison/maddison-project/home.htm 2013 version.
Figure 2 shows that from about 1960 until 1990 Australia was richer than the UK. After that, the growth dividend of Thatchernomics allowed the British to catch up again to the Australians.
Figure 3: British and New Zealand GDP per capita (1990 Int. GK$PPP), 1820 – 2010
Source: The Maddison-Project, http://www.ggdc.net/maddison/maddison-project/home.htm 2013 version.
Figure 3 shows that New Zealand was richer than the UK in the mid-20th century. The lost decades in New Zealand from 1974 to 1992 let the sick man of Europe overtake New Zealand before Thatchernomics caused a growth spurt in the UK to take the British well ahead.
Figure 4: Australian and New Zealand GDP per capita (1990 Int. GK$PPP), 1820 – 2010
Source: The Maddison-Project, http://www.ggdc.net/maddison/maddison-project/home.htm 2013 version.
Figure 4 shows that New Zealand was richer than Australian in the first part of the post-war period. The divergence started with the onset of the lost decades in New Zealand in the early 1970s.
Utopia, you are standing in it! 
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