A 1943 letter to the editor of the journal Nature suggests that the anglicized pronunciation of joule was accepted not only by many in the public, but by authorities for some time:
A century has passed since Joule read his paper on the relation between heat and work at the meeting of the British Association at Cork on August 26, 1843. It is unfortunate that a difference of opinion has arisen about the correct pronunciation of his name….
The “Oxford English Dictionary” gives dzaul as the pronunciation of the unit, where au represents the sound in the word loud. In “Webster’s New International Dictionary” (1911) is found the statement:
“The proper name is pronounced joul (ou as in out) and this is the correct pronunciation for the unit; but the incorrect jool [oo as in food] is very common especially in the United States”.
In “Chambers’s Technical Dictionary” (first published 1940) we find the contradictory statements
joule, jool (Phys.). A unit of energy equal to 107 ergs. (Named after F. [sic] P. Joule, 1918–89 ; name pronounced jowl.)
Kenyon & Scott's A Pronouncing Dictionary of American English, published a few years later, provides multiple pronunciations:
Joule: dʒaʊl, dʒul, dʒol | joule dʒaʊl, dʒul
As Pauling was born in 1901, it seems plausible that he learned to pronounce joule to rhyme with jowl, and that this pronunciation would have been accepted in some quarters— if not by all. People tend to pronounce names and terms as they learn then, without revisiting them, and that is true of scientists as much as anyone else.
A 2008 paper by JR Schwyter, Setting a standard: Early BBC language policy and the Advisory Committee on Spoken English, reports that "the Committee received 27 letters relating to the pronunciation of the last name Joule, as in James Prescott Joule, the scientist (19 January 1933)… In the light of such correspondence, the Committee (30 November 1933) decided on the pronunciation… ‘JOOL’*.
Over the course of the twentieth century, it seems the "jewel" pronunciation came to predominate. Perhaps the BBC Radio influence helped. Perhaps it rose because it was the more prevalent American pronunciation (per Webster's as quoted above), and American physicists rose in both numbers and preeminence, especially in the nuclear age. Perhaps it is because, related to this, the American postwar tendency is not to anglicize pronunciations, and the fact that Joule was an Englishman may be lost on the public. It is hard to attribute shifts in pronunciation to any one cause or moment in history.
The current OED entry notes
Although some people of this name call themselves /dʒaʊl/ and others /dʒəʊl/ (D. Jones Everyman's Eng. Pronouncing Dict. (ed. 11, 1956), G. M. Miller BBC Pronouncing Dict. British Names (1971)), it is almost certain that J. P. Joule (and some at least of his relatives) used /dʒuːl/ . For evidence on this point see Nature (1943) CLII. 354, 418, 479, 602.
For what it's worth, this was the pronunciation given in its first edition—
— immediately after the entry for Joul(e, obs. form of JOWL.