With some exceptions, candidates up for election are usually natural born citizens that are native speakers of the government's language. With the exception of the first generation of leaders in Israel, when revived Hebrew was almost nobody's mother tongue, have any immigrants learned a local language as an adult well enough to be elected to a governmental office using that language?
Arnold Schwarzenegger is a native German speaker from Austria and a naturalized US citizen who became Governor of California. Wikipedia indicates that his grasp of English was quite weak in childhood. Schwarzenegger himself indicated that his command of English was weak in 1968. In 1968 he was 20 or 21 years old, indicating that his mastery of English came as an adult.
Martin Van Buren, the eighth president of the United States, was a native speaker of Dutch and learned English later. This case might not fit your requirements as he was a native-born US citizen from New York.
Thomas Paine was twice elected in the French Convention during the French revolution. (I don't know how good his French was, and whether this was relevant).