The origins of this phrase are, I believe, London Cockney slang, when criminals would consider someone who informed on their activities to police as “a grass”. Hence, “grassed up” entering vernacular over here. It is akin to “ratted out” in the US, as you point out.

Also, from Google: “The origin derives from rhyming slang: grasshopper — copper; a ‘grass’ or ‘grasser’ tells the ‘copper’ or policeman.” That comes only a few years after the term grass was coined and there seems little reason to doubt it as the derivation.”

Glad you enjoyed the article. :)