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)
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.