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.