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The Spanish might be the best possibility: "primo" for a male cousin; "prima" for a female cousin. According to the ONLINE ETYMOLOGY DICTIONARY (credible source? I don't know; neither my Oxford-American nor my Webster's has the etymology for this word), *cousin* is from Old French *cosin*, from Latin *consobrinus*, which means "mother's sister's child".

I've never said this aloud before, but here's one possible answer to the uncle/aunt part of the question:   An uncle or aunt is a special case of cousin, specifically, they're one's first cousins once removed (and so are nephews and nieces).   So I suppose you could fall back on that.   But no one would understand you, so it's hardly worth mentioning.