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The phrase cheats never prosper (also, earlier, cheating never thrives) means: dishonest or deceitful actions, while they may offer a short-term advantage, do not lead to long-term success or well-being.

The English folklorists Iona Opie (1923-2017) and Peter Opie (1918-1982) mentioned this phrase in The Lore and Language of Schoolchildren (Oxford University Press, 1959) [chapter 10: Unpopular Children: Jeers and Torments, page 182]:

Amongst London boys a cheat is generally referred to as a ‘wog’, sometimes a ‘clot’. The girls cry out ‘Cheats never prosper’ or just ‘cheater’.

The English author and translator John Harington (1560-1612) had expressed a similar idea in The Most Elegant and Witty Epigrams of Sir Iohn Harrington, Knight, Digested into Foure Bookes: Three whereof neuer before published (London: Printed by G. P. for Iohn Budge […], 1618) [Book 4, page unnumbered]:

5 Of Treason.
Treason doth neuer prosper, what’s the reason?
For if it prosper, none dare call it Treason.

These are, in chronological order, the earliest occurrences of the earlier phrase cheating never thrives that I have found:

1-: From Tusser Redivivus: Being Part of Mr. Thomas Tusser’s Five Hundred Points of Husbandry; Namely, For the Month of August. With Notes. To be Published Monthly. N° VIII. (London: Printed, and are to be sold by J. Morphew, 1710), by the English agricultural writer and poet Thomas Tusser (c. 1524-1580) and Daniel Hillman [page 5]—thyth is an obsolete form of the noun tithe:

Of the Tyth somewhat has been spoke in former Months, therefore the less will serve here. It is certain the Tyth is not the Farmers; and withholding it is Cheating, and Cheating never thrives.

2-: From A Tour in America, in 1798, 1799, and 1800. Exhibiting Sketches of Society and Manners, and a particular Account of the American System of Agriculture, with its recent Improvements (London: Printed for J. Harding […] and J. Murray […], 1805), by the English agriculturist Richard Parkinson (1748-1815) [Section 29, page 506]—Mr. Bell, of Philadelphia, is one of the “men who have risen from nothing”:

I found that the first money Mr. Bell acquired (what is called fortunate money, since all his great riches arose from it) was by buying damaged hats, that had been imported, and were rotten—a pretty large quantity of them for a very small sum of money. He used to sit up, and in the night, for fear any one should see him, to trim them for sale: had they been seen before dressing, the market would have been over. The method he adopted to make them saleable was a singular one: it was by oiling, and afterwards pressing them with a hot iron. He likewise told me several other stories similar to the preceding. It is a common saying in England, that “Cheating never thrives:” but, in America, with honest trading you cannot succeed.

3 & 4-:From The Island Bagatelle; Containing Poetical Enigmas on the Estates in each Parish in the Island of Grenada (Grenada: Printed by W. E. Baker, 1829), by the British author Frederick William Naylor Bayley (1808-1853):

3-: From The Spirit of Warning in the Land of Forgetfulness [stanzas 14 & 15, page 91]:

“And if, by chance, some modern gentleman
Shall introduce you to his pretty daughter,
And when, with sweet delight, your eyes may scan
O’er all the airs and graces he has taught her,
In all your corps, let not one single man,
Or try to catch, or keep her when he’s caught her;
Or elseway make him marry—on your lives,
’T is an old adage—Cheating never thrives;”

“Or, if it thrive at all, it thrives the least
Upon a soldier,—and ’t were better far
That officers in their mal-doings ceased,
And gave themselves to virtue and to war.”

4-: From The Spirit of Warning in the Land of Forgetfulness [stanza 20, page 93]:

“And now,” the spirit said, “I steer my course
To sons of med’cine—curers of disease—
And many doctors I shall have to curse,
And, therefore, many doctors to displease:
The hospitals are daily growing worse,
The patient’s med’cine bad, tho’ mix’d with ease.
Doctors, be warn’d—I tell you, on your lives,
Do better deeds—for, ‘Cheating never thrives.’”

These are, in chronological order, the earliest occurrences of the phrase cheats never prosper that I have found:

1-: From Boys of English History. VIII.—Lambert Simnel, the baker’s boy who pretended to be a king, published in The Boy’s Own Paper (London, England) of Saturday 21^st June 1879 [page 355, column 3]:

He would go on to tell how all this was because designing men had put into his head foolish ambitions, and taught him to repeat a likely-looking story. And if one had questioned him further, doubtless he would have confessed that he was happier far now as a humble turnspit than ever he had been as a sham king, and would have warned one sadly that cheats never prosper, however successful they may seem for a time; and that contentment with one’s lot, humble though it be, brings with it rewards infinitely greater than riches or power wrongly acquired.

2-: From Enigmatic Parallelisms of the Canton Dialect, published in The China Review: Or, Notes and Queries on the Far East (Hong Kong, China) of June 1886 [Vol. 14, No. 6, page 360, column 1]:

False balances, though common enough, vary very slightly from the true. Some of the shops are said to have two sets of balance weights, one for weighing in and the other for weighing out money. But this is looked upon as a dangerous practice and the Chinese say as we do that ‘cheats never prosper.’