Cop is a shortened form of “copper” and was in use in the UK before it was in the US. It stems from the slang verb “to cop”. Cop is used in various combinations such as “cop out” (evading one’s responsibilities), “not much cop” (of little use) and “it’s a fair cop” (you’ve caught me bang to rights). The last example connects with the custodian of the law, or peeler (same derivation as bobby) and the derivation of the verb “to cop” in this sense is obscure but may be related to the Old French verb “caper” meaning to seize. (Thank you, OED!)

Edit: Drat, FF, you beat me to it with a really good link!

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