We don't have rhyming slang in the USA, which makes the standard explanation (Cockneys trying to throw off the cops) seem quite plausible. Americans have developed their own slang for criminal or questionable activities, but rhyming slang appears to have developed in London during the mid-1800s and never crossed the pond this direction.
That it exists in Oz strengthens the "cockney criminal" theory, since an entire shitload of Cockney criminals were shipped to Botany Bay on a regular basis to form the workforce of Aussie colonies and make the Poms feel better about prosecuting people (and kids) for crimes such as stealing a loaf of bread. "It's not like we killed the lad, after all, we gave him a chance!"
Slang exists in every language, and seems to thrive among the lower classes, most likely because they are likely to be under the eye of the police. Some American slang is a holdover from British slang, such as "cop" or "copper" for the police, who wore uniforms with copper buttons in the UK. But most of our slang has come from the lower classes, with an amazing amount co-opted from American Blacks. One of my favorite factoids is that both "Jazz" and "Rock-and-roll" began among Black Americans as slang terms for the sex act.
The main reason the Poms had for establishing the Botany Bay penal colony was that the American Revolution had closed off their previous dumping ground for the overflow of criminals from British prisons; I wonder if any of the uniquely American slang you take for granted originated with earlier thieves' cants. Had rhyming slang developed a century earlier than it did, you seppos would probably use it too.
I am doubtful about the word copper having anything to do with uniform buttons, which would have been brass; more plausibly it is a corruption of (or shares a latin common root with) 'capture'.
'to cop' meaning 'to take' is common usage in the UK - for example 'cop a butchers at that' meaning 'take a look at that'; or 'cop a feel', meaning 'take a feel' (usually in reference to the breasts of one's chosen sexual partner, and with a vague overtone of the absence of consent 'She was pissed as a newt*, so I took the opportunity to cop a feel'). One who takes would therefore be a 'Copper' - although to my knowledge, nobody refers to the one taken as a 'Copee'
*I have no idea why aquatic amphibians of the family Salamandridae are chosen to epitomise intoxication; nor why Americans are enraged when they should be intoxicated.
