How and when did the ('chiefly British') sense of 'wheeze', meaning 'good plan, scheme' arise?
The American Heritage Dictionary entry for this sense of wheeze adds only a [chiefly British] caveat:
wheeze ... [noun]
...
- [chiefly British] A clever scheme.
Collins adds a [slang] caveat:
wheeze [noun]: ...
- [slang] [Brit] a trick, idea, or plan (esp in the phrase good wheeze)
But neither these dictionaries nor RHK Webster's, Wiktionary, CED, Lexico nor the Online Etymological Dictionary suggests how the 'scheme' sense evolved from the original 'breathe raspingly' sense.
Those non-paywalled online dictionaries listing the 'scheme' sense always include it as a polyseme, not a homonym. It is possible that the '[informal] a hackneyed joke or anecdote' sense (Collins) is an intermediate, though Lexico says that this is mainly an American usage. But I have found not even a suggestion of how or when the 'scheme' sense, I'd say mainly a dated [CED] UK usage and very informal, with a strong connotation of roguishness, arose.
Solution 1:
Apart from a few early Australian usages, the term is mainly used in BrE according to GDoS. The term appears to be from late 19th century and appears to have developed out of its original sense of “theatrical gag” as suggested by A Dictionary of Slang and Unconventional English by Eric Partridge
Wheeze:
- a trick or dodge frequently used.
1885 [Aus] Bulletin (Sydney) 14 Feb. 7/4: And we remember on one occasion, when some miscreant cracked this sorry wheeze, a gentleman, whom Melancholy had Marked for Her Own, said it was ‘almost good enough for Punch.’
1890 [UK] Sporting Times 1 Feb. 1/3: [He] retired to the Woods, to see if the Forty Bank Notes were Still Between the Leaves, and Cogitate upon the Success with which his B.C. 55 Wheeze had Worked.
1904 [Aus] Sun. Times (Perth) 13 Mar. 11/3: A schemester who works every wheeze.
1908 [UK] A.N. Lyons Arthur’s 22: That ’ere was a champion wheeze while it lasted.
................
2000 [UK]  Guardian G2 18 Jan. 14: It is a good wheeze.
Solution 2:
The OED gives an earlier date and a more precise origin:
Wheeze (n.) 2. Originally Theatre slang, A joke or comic gag introduced into the performance of a piece by a clown or comedian, esp. a comic phrase or saying introduced repeatedly; hence, (gen. slang or colloquial) a catchphrase constantly repeated; more widely, a trick or dodge frequently used; also, a piece of special information, a ‘tip’.
1864 J. G. Bertram Glimpses Real Life 131 The art of getting up ‘wheezes’, as the clown's jokes are called.