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Quite right David! I knew I should have checked before making that claim. It's Worsley's log that is at Cantabury, but not digitized for public viewing. Here's another page of Scott Polar Research Institute exhibit of the sextant, actually owned by Huberht Hudson. https://www.spri.cam.ac.uk/museum/shackleton/expeditions/endurance/

This from Worsley himself: “After a few days of this particular type of navigation, however, I became so adept at it that I got some surprisingly correct results. On the rare occasions when I got an observation of the sun I had to kneel on the thwart to use my sextant, with two seamen holding me up, one on each side, gripping me tightly to prevent the violent motion of the boat heaving me bodily overboard, sextant at all. As I made the observation Shackleton, beneath the canvas covering of the boat, would take the time by the chronometer. This chronometer, by the way, was the sole survivor in working order of 24 with which the Endurance had left England two years previously. It was a valuable one, and a few months after the boat journey, when I was boastfully showing it to friends, Shackleton said jokingly,

  ‘You're very proud of the old chronometer, Skipper. Would you like to have it?’

  ‘Rather’, I answered and with typical generosity he said, ‘Very well. It's yours.’ (Worsley, Endurance: An Epic of Polar Adventure, p. 110)."

This chronometer is also in the SPRI along with a nice short film: https://www.spri.cam.ac.uk/museum/shackleton/expeditions/endurance/

Another boxed chronometer apparently survived and is in the National Martime Museum in Greenwich.

A most wonderful account of the Navigation of the James Caird on the Shackleton Expedition can be found in Lars Bergman, Robin Stuart, et al. 2018 paper, including transcripts of Worsley’s log and notebooks. Great stuff!!