“There was an Old Man who said, ‘How
Shall I flee from that horrible cow?
I will sit on this stile,
And continue to smile,
Which may soften the heart of that cow.'”

Lear illustrates with with a charmingly grotesque picture of the Old Man, his arms flung backward, grinning at the cow.
–
Thank-you. Added to my note. -RML

Not at all, nitpicks are welcome, and hopefully fixed.

Thanks! I can say with certainty that the only occurrence of “camel dung” among Churchill’s 20 million known words is his 23Oct43 question to Eden’s Undersecretary, Alexander Cadogan, as to whether the Turkish word AMGOT “means camel dung or something equally unpleasant?” (AMGOT was the acronym for “Allied Military Government for Occupied Territories.”)

Cadogan replied: “Prime Minister, I have established, on high authority, that AMGOT does not, in Turkish, signify camel dung. It does not correspond to any single Turkish word. There are, however, two Turkish words, ‘Ahm’ and ‘Kot,’ which an English scholar would, not incorrectly, translate as ‘Cunt’ and ‘Arse’…. The Turkish term for camel’s dung is deve gübresi.” (Cadogan to his Diary: “My task is complicated by the PM suggesting—or sending—spasmodic replies to telegrams which really require some modification.”)

Sorry, couldn’t resist.

There is one hearsay report that he said something similar to the words you quote, but the evidence is not strong, in fact contradictory. The context was Churchill’s criticism of the 1939 Palestine White Paper, which he regarded as anti-Semitic. It’s in Michael Makovsky’s excellent book, Churchill’s Promised Land, pp. 168-69: but you need to read it in full:

As much as Churchill’s respect and sympathy for the Jews had strengthened, his opinion of the Palestinian Arabs and their cause had worsened. He could not understand why the Arabs were indulged when they, in contrast to the Jews, reaped more than history owed. He argued that “elsewhere over vast regions inhabited by the Arabs independent Arab Kingdoms and principalities have come into being such as had never been known in Arab history before.” More specifically, he did not understand why the government favored the Palestinian Arabs who historically had been hostile to Britain and its interests (fighting for the Ottomans in the First World War, for instance), and still were. He expressed great frustration with the perverse strategic choices made by the government of favoring enemies over allies. He exclaimed, “We are now asked to submit—and this is what rankles most with me—to an agitation which is fed with foreign money and ceaselessly inflamed by Nazi and by Fascist propaganda.” Perhaps reflecting his complete exasperation with the Palestinian Arabs, in 1938 he got into a big argument with Malcolm MacDonald, the new colonial secretary, in the lobby of the House of Commons. MacDonald recalled the argument several decades later: “He told me I was crazy to help the Arabs, because they were a backward people who ate nothing but camel dung.” While these might not have been Churchill’s exact words, the gist of the comment jibed with what he had thought of the Palestinian Arabs at least since encountering them in the early 1920s.