I have researched this topic as well. I would like to point out a few aspects to this discussion that may be useful. First of all, it is correct (and Wikipedia says so as well) that today's Australian publications, including newspapers and digital media, almost without exception (except for certain proper nouns such as the Labor party and Victor Harbor) follow the UK norm of "-our" endings. It is correct that several Australian publications in the 20th century preferred the "-or" endings. Newspapers such as "The Age" from Melbourne and "The Advertiser" from Adelaide used the American "-or" spellings until the middle of the 1990's. They changed to British spellings in the second half of the 1990's probably with the advent of digitised and computerised publications. I think a similar trend occurred in Canadian newspapers as well.
I should add, though, that it is probably not accurate to claim that Australian spellings "changed" from American to UK norms. This gives the spurious impression that American spellings used to be preferred to UK spellings in the early 20th century. That seems to have never been the case as I argue below.
First of all, even when "-or" endings were more conspicuous in Australia, they were certainly not exclusive and co-existed with "-our" endings. For a Victor "Harbor" in South Australia, spelt without the "u", there was a Sydney "Harbour" bridge that was always spelt with the "u". Also, the "-or" spellings were probably the ONLY instances of American spelling norms co-existing in Australian publications. The more conspicuous differences such as "realise", "organise", "analyse" were always spelt using British/Commonwealth norms instead of the American style of ending with "-ze", as were "theatre" and "centre" or even "manoeuvre" (American spellings prefer "-er" endings), as also were "modelling" and "travelling" (American english prefers single "l"). It is, therefore, not correct to say (only on the presence of "-or" endings) that Australian spellings used to be closer to American spellings in the past. That seems never to have been the case. In fact, in spite of the use of "-or" endings, overall Australian spelling norms have always been closer to UK norms.
This, in my mind, also belies the argument that the prevalence of American dictionaries were the cause of American style spellings. I think the "-se/-ze" differences between American and British spellings are much more frequently seen in writings across the world than the "-our/-or" endings. Had American dictionaries been that influential in Australia, then the "-ze" endings (and the "-er" endings) would have likely been adopted in Australia as well. They never were.
To my mind, the most logical reason for the almost universal acceptance of British spelling norms in today's Australian English, including the apparent abandonment of the "-or" endings, took place because the "-or" endings, though seen more often in the early 20th-century than now, were never dominant and were still considered somewhat anomalous. With spell-checkers and computerised publishing, it would be easier to follow one norm, either UK or US, for spellings. The UK spellings were adopted probably because they were always more prevalent. Today, the only conspicuous American spelling in Australian media comes from the Australian "Labor" Party due to historical and political reasons (although "labour" with the "u" is the preferred spelling in all other contexts), and the use of the spelling "program" instead of "programme" in all contexts. Note that the Democratic Labo(u)r Party in Australia, that used to also spell "Labor" recently switched back to the UK spelling with the interesting comment: "Putting (YOU) back in Labo(u)r!" Perhaps the Australian Labor Party will also bring the "u" back in Labor in the not so distant future.
Finally, the near future of Australian spellings seems to be much more closely aligned with British norms. UK spellings are taught in schools, and in spite of Americanisms pervading all aspects of Australian society, including language, Australians probably will likely not change to American spellings because doing so may appear as a new type of cultural colonialism. If Canada, the next-door neighbour of the US, retains mostly UK spellings, it is difficult to see how and why Australia would switch to US spellings. Funny that all the commonwealth nations (read India, Pakistan, Singapore, Hong Kong) have taken to shunning so many practices of their colonisers (the British), but have all retained their spellings!
Best Answer
According to Oxford English Dictionary, the answer is yes. Zinke was the earliest attested spelling of the word, with zink also preceding zinc.
The earliest attested use of the word is from 1651.
The spelling you suggested is believed from 1734.
And finally, the earliest attested spelling with a 'c' is from 1813
You ask: When might it have ceased to be used?
OED indicates that zink was used into the 1800's, in its remarks on the forms of the word:
However, jlovegren proved that some uses from the 1900's can be found. I would add to that these snippets from 1906 and 1914, respectively. However, the dates provided by OED are probably an accurate measure of when each form was in common use.