New South Wales The earliest is this New South Wales Stamp, 4d of the 1888 centennial of settlement issue. Its origin is discussed here.

Other early and first country issue Cook stamps include:

Landing of Cook. 3d, (1st. Voyage, Poverty Bay, October 8th, 1769). It is not based on any contemporary picture.

The designer of this series of stamps was L.J. Steele - a genre painter - best remembered today for his joint effort with Goldie  - the Arrival of the Maoris in New Zealand - (See - Figure 3). 
This series was hugely controversial at the time because it was New Zealand's first commemorative series. The one earlier commemorative stamp - about the Boer War - was hidden within the first pictorial set.  Philatelists then saw commemorative stamps as the road to ruin - and they had a point didn't they? The designs were also panned and one can see why. Still they have a high catalogue value today. Catalogue

It is not based on any contemporary picture.

Whitianga in Mercury Bay has a Cook Museum.

* An authoritative source says that the stamp depicts Cook's "annexation". Cook's instructions were only to take possession with the consent of the native people. He certainly did not have that. His act of annexation in Mercury Bay was not given effect to and today is treated as a footnote to the colonisation of New Zealand. The effective annexation was that of Hobson on May 21st 1840, in Auckland, after the earlier signing of the Treaty of Waitangi and its hawking around as many Maori chiefs as would sign, by Major Bunbury. Another source says that is what is depicted. From the content showing a shore party I think the former is correct.

On 2c Washington, 

On 5c Roosevelt. 

Rarotonga (Cook Islands) (and others)  1932 Cook definitive.

Not identified as Cook on the stamp but clearly is derived from Dance, but with the hat added and different uniform.  Paul den Ouden kindly advised on the origin " This was drawn from a photograph of a steel engraving of the famous painting by N. A. Dance, R.A., Mitchell adding the admiral's hat. Another steel engraving from the same portrait was used for the 1½d. value of the 1920 set. The two engravings, although of the same painting, varied in the expression on Cook's face and other details. The Mitchell referred to is L.C. Mitchell, of Wellington, NZ who was the designer of the complete 1932 set. I've attached a scan of one of the essays, sans hat. "

New Zealand   1935, Landing of Cook 2/-  (This one an Official stamp - used by Government departments.)

New Zealand 1959. 2d Marlborough Centennial. Careening of Endeavour at Ship Cove

The first  stamp just about Endeavour.  It is not based on any contemporary picture.  Ship Cove is in the Marlborough Sounds at the north end of the South Island. Other stamps above have a ship in the background, which from context is Endeavour but it is not identified on the stamp as such. The first identifying Endeavour is the 1940 New Zealand issue.

Niue   1950  The first stamp to identify Resolution.

Australia 1964, Navigators 7/6 This short lived issue was replaced in 1966 by a decimal stamp of 75c denomination, using the same design.