Hi there,
Do you think he may have gone back to England 1850, and then perhaps returned to NSW when news of the gold discoveries reached the UK ... perhaps he travelled back and forth as a Surgeon as part of the crew on various ships carrying the gold seekers. The population of Australia escalated

with various episodes of 'gold fever' causing masses to flock to many different mining sites in NSW and Victoria in the 1850's.
Do you have these cuttings for John Cates
Sydney Morning Herald (SMH) 1 February 1843 has him in Sydney (Town)
SMH 20 November 1848 has he and his wife at Parramatta in Sept 1848
Sorry, but I can't be positive that he was in NSW in 1858 or 1860 when the NSW Medical Board listed him as being at Sydney ... Have you contacted the NSW Medical Board to see if they have archive records ? the contact details are at their "about us" page on this link
http://www.mcnsw.org.au/ Cheers, JM