Meaning
Speaking in an affected, over-refined 'proper' English accent to sound educated or upper-class — often used mockingly, of someone who puts on airs, abandons their natural Patois, and strains to appear posh. The result can sound stilted and pretentious, and 'speaky-spokey' pokes fun at the pretender caught between two ways of talking, neither natural nor convincingly grand. It belongs to the sharp Jamaican vocabulary of class and pretension, ridiculing those who forsake their own tongue to ape the speech of the well-to-do.
Examples
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Since she come from farrin she a chat all speaky-spokey.
Since she came back from abroad she talks all posh and affected.
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Him get di office job an turn speaky-spokey.
He got the office job and started putting on a refined accent.
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Nuh badda wid di speaky-spokey, chat normal.
Don't bother with the fancy accent, talk normally.
Origin
A playful reduplication mimicking stilted 'speak/spoke' speech — mocking the affected, sing-song 'proper' accent of one trying to sound refined.
How it’s used
Usually mocking, aimed at pretension and putting on airs. Compare 'dry-land tourist', 'hitey-titey'.