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Dunthorne published his tables (“The Practical Astronomy of the Moon: or, New Tables of the Moon's Motion") in 1739, claiming that they were constructed from Newton's theory. In the next sentence he admits, though, that "the Newtonian numbers are a bit deficient". So the first useful ephemerides of the moon were calculated by Tobias Mayer less than two decades later. They were reviewed by Bradley, the Astromomer Royal at the time, and served as the basis of the "Nautical Almanac" published by Nevil Maskelyne in 1766.

Dunthorne played only a minor role in the history of lunar distances as he was employed by Maskelyne as "Comparer" in the calculation of the NA for some years. And his method for clearing the distance was included in the "Tables Requisite..." also published by Maskelyne in 1766 - alongside with another method for clearing the distance by Israel Lyons. They both received 50 Pounds by the Board of Longitude for their work; the Board also paid Dunthorne for his work as "Comparer". It wasn’t the first method for clearing the lunar distances, as the problem had been discussed since the 17^th century – a useful list of (mainly french) contributions can be found in Guy Boistel’s “L’astronomie nautique au XVIIIème siècle en France”, p. 643-644.

For Newton's minor role in the development of the lunar distance method you can consult Steven Wepster's dissertation "Between Theory and Observation" (New York 2012).

Frank is right, of course, that lunar distances were widely used by navigators until the mid-19th century. Later the method was still used for chronometer control until radio time signals became available. By the time of Littlehales - 20th century by then - they were of course outdated, but still used by some - Joshua Slocum in his circumnavigation for instance. "Finding Longitude" by Richard Dunn and Rebekah Higgitt tells the story very well; for anybody who needs something shorter I would recommend "Man is not lost. A record of two hundred years of astronomical navigation with "The Nautical Almanac" 1767 - 1967".