While Russian doesn't have definite/indefinite articles, the word order can inform on the degree of knowledge participants to a conversation have about the things they're talking about. Emphasis, meaning new ideas/elements, goes to the end of the sentence, while things that were previously known of fall back towards the beginning.
The definite article "the" means that it is already known which pencils are being discussed, so "He does not have the pencils." would be translated as «Карандашей у него нет.»
Strange that "hasn't any" is exclusive to British English! I've been speaking American English all my life, and it was the first thing that popped into my mind. I have spent two or three years in England over my life, though, so maybe it has crept into my usage from that route.